Hybrid connections on Hessian manifolds

TL;DR

This paper introduces hybrid connections on Hessian manifolds, exploring their properties, and constructs a new natural connection on the unit ball that bridges hyperbolic models.

Contribution

It defines hybrid connections with specific geometric properties, characterizes their differences from the Levi-Civita connection, and identifies a canonical model in pseudo-Euclidean spaces.

Findings

The difference between hybrid and Levi-Civita connections is given by the logarithmic differential of a Hessian potential.

A new natural connection on the open unit ball is constructed, bridging Cayley-Klein and Poincaré models.

A unique pseudo-Riemannian metric makes unparameterized geodesics have constant speed with respect to the isochrone metric.

Abstract

A Hessian manifold $(M,D,g)$ is a manifold $M$ with a flat connection $D$ and a Riemannian or pseudo-Riemannian metric $g$ that is locally of the form $D^2 f$ for some function $f$. On a Hessian manifold $(M,D,g)$, we define a hybrid connection as an incompressible affine connection $\nabla$ that is projectively flat relative to $D$ (its unparametrized geodesics are aligned with the affine structure of $D$) and whose first-order infinitesimal holonomy at each point of $M$ is an infinitesimal isometry of the pseudo-Riemannian metric $g$. In this paper, we investigate the properties of hybrid connections, proving in particular that for a hybrid connection $\nabla$, the difference $\nabla-D$ is determined by the logarithmic differential of a function that serves as a Hessian potential for $g$. In the special case of pseudo-Euclidean manifolds, we identify canonical models and obtain in particular a new natural connection on the open unit ball that provides a compromise between properties of Cayley-Klein and Poincar\'e hyperbolic models. We also find a unique (up to a scaling) pseudo-Riemannian metric $h$ such that unparameterized geodesics of $\nabla$ have a constant speed with respect to the so-called isochrone metric $h$.

Tables

Table 1. Table 1. Comparison of natural connections for the unit open disk. For readability we adopt coordinates ( x , y ) 𝑥 𝑦 (x,y) instead of ( x 1 , x 2 ) subscript 𝑥 1 subscript 𝑥 2 (x_{1},x_{2}) .

Connection	Flat	Cayley-Klein	Poincaré	${𝒮}_{-1}^{-}(2,0)$
Preserved metric	$d{x}^2+d{y}^2$	$\frac{d{x}^2+d{y}^2}{1-x^2-y^2}+\frac{{\left(xdx+ydy\right)}^2}{{\left(1-x^2-y^2\right)}^2}$	$\frac{4\left(d{x}^2+d{y}^2\right)}{{\left(1-x^2-y^2\right)}^2}$	$\times$
Preserved volume form	$dx\wedge dy$	$\frac{dx\wedge dy}{{\left(1-x^2-y^2\right)}^{3/2}}$	$\frac{4dx\wedge dy}{{\left(1-x^2-y^2\right)}^2}$	$\frac{dx\wedge dy}{{\left(1-x^2-y^2\right)}^3}$
Curvature form	0	$\frac{-dx\wedge dy}{{\left(1-x^2-y^2\right)}^{3/2}}$	$\frac{-4dx\wedge dy}{{\left(1-x^2-y^2\right)}^2}$	$\frac{-2dx\wedge dy}{1-x^2-y^2}$
Geodesically complete	$\times$	✓	✓	✓
Infinitesimally conformal	✓	$\times$	✓	✓
Straight geodesics	✓	✓	$\times$	✓

Equations

$${\partial }_{X}Y=\sum _i{X}_i\frac{\partial Y}{\partial x_i}=\sum _{ij}{X}_i\frac{\partial Y_j}{\partial x_i}{e}_j.$$

$${\nabla }_{X}Y={\partial }_{X}Y+\Gamma (X,Y)$$

$$\Gamma (X,Y)=\sum _{ijk}{\Gamma }_{jk}^i{X}_j{Y}_k{e}_i$$

$${\nabla }_{X}Y={\partial }_{X}Y.$$

$${\nabla }_{X}Y=0$$

$$\dot{v}(t)=-\Gamma\{\gamma(t)\}\big{(}\dot{\gamma}(t),v(t)\big{)}$$

$$\dot{v}=-\Gamma\{\gamma\}\big{(}\dot{\gamma},v\big{)}.$$

$$\ddot{\gamma}=-\Gamma\{\gamma\}\big{(}\dot{\gamma},\dot{\gamma}).$$

$$\forall i,j,k,{\Gamma }_{jk}^i={\Gamma }_{kj}^i.$$

$$R_{ijk}^l={\partial }_i{\Gamma }_{jk}^l-{\partial }_j{\Gamma }_{ik}^l+\sum _p\left({\Gamma }_{ip}^l{\Gamma }_{jk}^p-{\Gamma }_{jp}^l{\Gamma }_{ik}^p\right)$$

$$M(i,j)$$

$$\tilde{\Gamma }\{\varphi \}\circ (L,L)=L\circ \Gamma -H$$

$$\tilde{\Gamma }\{L\}\circ (L,L)=L\circ \Gamma .$$

$$\Gamma =L^{-1}\circ H.$$

$$\Gamma (u,v)=a(u)v+a(v)u$$

$$a(u)=-\frac{F^{\prime }(u)}{F(x)}$$

$$\forall i\ne j,{\Gamma }_{ii}^i$$

$$\forall i,j,\forall k\notin \{i,j\},0$$

$$\forall v,\Gamma (v,v)\text{ is colinear to }v.$$

$${\Gamma }_{ii}+\lambda ({\Gamma }_{ij}+{\Gamma }_{ji})+{\lambda }^2{\Gamma }_{jj}$$

$${\Gamma }_{ii}^k+\lambda ({\Gamma }_{ij}^k+{\Gamma }_{ji}^k)+{\lambda }^2{\Gamma }_{jj}^k=0$$

$${\Gamma }_{ii}^j+\lambda ({\Gamma }_{ij}^j+{\Gamma }_{ji}^j)+{\lambda }^2{\Gamma }_{jj}^j=\lambda \left({\Gamma }_{ii}^i+\lambda ({\Gamma }_{ij}^i+{\Gamma }_{ji}^i)+{\lambda }^2{\Gamma }_{jj}^i\right)$$

$$\forall v=\sum x_i{e}_i,\Gamma (v,v)=\left(\sum {\Gamma }_{ii}^i{x}_i\right)v.$$

$$\Gamma (u,v)=a(v)u+a(u)v.$$

$${\Gamma }_{ij}^k={\delta }_{ik}{a}_j+{\delta }_{jk}{a}_i$$

$$R_{ijk}^l={\delta }_{il}(a_j{a}_k-{\partial }_j{a}_k)-{\delta }_{jl}(a_i{a}_k-{\partial }_i{a}_k)+{\delta }_{kl}({\partial }_i{a}_j-{\partial }_j{a}_i).$$

$$R_{ijk}^l={\partial }_i{\Gamma }_{jk}^l-{\partial }_j{\Gamma }_{ik}^l+\sum _p\left({\Gamma }_{ip}^l{\Gamma }_{jk}^p-{\Gamma }_{jp}^l{\Gamma }_{ik}^p\right).$$

$$({\delta }_{il}{\delta }_{jp}-{\delta }_{jl}{\delta }_{ip})a_p{a}_k+({\delta }_{jp}{a}_i-{\delta }_{ip}{a}_j){\delta }_{pl}{a}_k+({\delta }_{il}{a}_j-{\delta }_{jl}{a}_i){\delta }_{kp}{a}_p.$$

$$\sum _p{\Gamma }_{ip}^l{\Gamma }_{jk}^p-{\Gamma }_{jp}^l{\Gamma }_{ik}^p={\delta }_{il}{a}_j{a}_k-{\delta }_{jl}{a}_i{a}_k.$$

$$a=\sum a_{i}d{x}_i$$

$${\mathrm{Ric}}_{jk}=\sum _i{R}_{ijk}^i.$$

Full text

In this paper, we give a complete classification of stiff connections in a local chart, identify canonical models and start investigating the global geometry of (similarity) pseudo-Euclidean manifolds endowed with a stiff connection.

In the conformal class of the pseudo-Euclidean metric $g$, a stiff connection $\nabla$ defines a pseudo-Riemannian metric $h$ such that unparameterized geodesics of $\nabla$ coincide with unparameterized geodesics of $g$ but have a constant speed with respect to the so-called isochrone metric $h$. In particular, we obtain a new natural connection on the open unit ball that provides a compromise between properties of Cayley-Klein and Poincaré hyperbolic models.

1. 1 Introduction

2. 1.1 Organization of the paper

3. 2 Reminder on affine connections

4. 3 Tensor formulas for projective charts

5. 3.1 The associated form

6. 3.2 Ricci tensor and incompressible connections in projective charts

7. 3.3 Affine change of variable

8. 3.4 Projective change of variable

9. 3.5 Restriction to an affine subspace

10. 3.6 Geodesics

11. 4 Classification of stiff connections

12. 4.1 Infinitesimal holonomy and coefficients of the curvature tensor

13. 4.2 Stiff connections

14. 4.3 Rigidity in dimension $d\geq 3$

15. 4.4 Flexibility in dimension two

16. 4.5 Canonical models of stiff connections

17. 4.6 Slicing canonical models and the sign of curvature

18. 4.7 Local isomorphisms between canonical models

19. 4.8 Zooming in and out

20. 5 Intermezzo: conformal considerations

21. 5.1 Infinitesimal homotheties

22. 5.2 Conformality for projectively flattened connections

23. 6 Parallel transport in canonical models

24. 6.1 The equipotential and radial foliations

25. 6.2 Parallel transport along rays

26. 6.3 Parallel transport along equipotential arcs

27. 6.4 Local holonomy

28. 7 A tale of three metrics (and five volume forms)

29. 7.1 Isochrone metric

30. 7.2 More about the isochrone metric

31. 7.3 Forms and curvature

32. 7.4 Parameterization of geodesics

33. 7.5 Restriction/Extension

34. 7.6 Example

35. 8 Global geometry of stiff connections on manifolds

36. 8.1 Developing map

37. 8.2 Deducing the pseudo-Euclidean structure from the connection

38. 8.3 Stiff connections on compact manifolds

39. 8.4 Geodesically complete stiff connections

40. 8.5 Geodesically complete stiff and weakly stiff connections on the disk

Note: Unless explicitly stated, all manifolds and pseudo-Euclidean spaces considered in this article are assumed of dimension $d\geq 2$.

1. Introduction

In this paper, we investigate a form of compatibility between projective and conformal flatness for affine connections.

As a motivation, note that planar charts of non-Euclidean surfaces are always unsatisfactory in one way or another, not only because the Gaussian curvature prevents the chart to map the metric to the Euclidean one.

For example, there are two famous representations of the hyperbolic plane as a disk: the Poincaré model and the Beltrami-Klein model. The first one is conformal (angles between curves are identical for the hyperbolic metric and for the Euclidean one) but hyperbolic geodesics appear as curved lines. In the second model, the geodesics of hyperbolic geometry are represented as the chords of the disk but the Euclidean angle at their intersection does not faithfully represent the true one.

In a similar way, gnomonic projection of the sphere sends great circles to straight lines, but it is not a conformal representation either.

To a Riemannian metric is associated the Levi-Civita connection, which is a symmetric affine connection, from which one can recover the metric up to multiplication by a constant. By focusing on the connection rather that the metric, one gets more flexibility and we will exhibit in particular an interesting family of connection of the Euclidean space whose geodesics are straight lines, and whose first-order infinitesimal holonomies are isometries.

It is still too much to ask for a connection on a subset of Euclidean space that would both have its geodesics straight, and be conformal, i.e. whose parallel transport between tangent spaces consists only in similarities.111By similarity of a pseudo-Euclidean space $E$, we mean the composition of a homothety and of an isometry. Note that a similarity multiplies the underlying quadratic form $q$ by a positive constant. This excludes the map $(x_{1},x_{2})\mapsto(-x_{2},x_{1})$ in ${\mathbb{R}}^{2}$ endowed with $q(x)=x_{1}^{2}-x_{2}^{2}$, which multiplies $q$ by $-1$. Indeed, we will see in Section 5.2 that such a connection must be trivial. Even if we relax the condition and only ask that, locally, the holonomy groups are contained in the group of similarities, then the connection must be flat, and more precisely it is the image of the trivial connection of ${\mathbb{R}}^{d}$ by a projective transformation.

This is why we relax in this article the conformality condition into a infinitesimal version. In order to work in a more general setting, we fix a signature $(p,d-p)$ and we will consider pseudo-Euclidean manifolds and similarity pseudo-Euclidean manifolds. These are manifolds $M$ with an atlas taking values in $d$-dimensional pseudo-Euclidean spaces of signature $(p,d-p)$, and whose change of charts consists in (restrictions of) isometries in the first case, and (restrictions of) similarities in the second case. In the first case the charts induce on $M$ a pseudo-Riemannian structure of signature $(p,d-p)$.

Given a pair $(M,\nabla)$ of such a manifold, of dimension $d\geq 2$, and of an affine connection on it, we say that $\nabla$ is stiff if

• •

$\nabla$ is symmetric,

• •

geodesics of $\nabla$ are straight lines in charts of $M$,

• •

the first-order infinitesimal holonomies of $\nabla$ are pseudo-Euclidean isometries in charts.

By charts we mean pseudo-Euclidean charts. Such a connection is in particular incompressible, i.e. locally its parallel transport preserves some volume form.

Note that, though geodesics are straight lines in such charts, we do not require that they have constant vector speed in these charts.

The setting of (similarity) pseudo-Euclidean manifolds is natural because any local transformation of pseudo-Euclidean space that preserves straight lines and is conformal must be a similarity.

The infinitesimal conformality hypothesis proves to be flexible enough to obtain an interesting family of connections that can be thought of as non-trivial deformations of the trivial connection. The main result of the paper is a complete local classification of stiff connections: for each type of pseudo-Euclidean space $E^{p,d-p}$ of signature $(p,d-p)$, we define between four and six canonical models $\mathcal{S}_{\lambda}^{\pm}(p,d-p)$ depending on the choice of a sign and a choice of constant $\lambda\in\{{-1,0,1\}}$.

We will also explore variations of these questions where we relax the incompressibility hypothesis. In Section 4, we prove that infinitesimal conformality implies that $\nabla$ is incompressible provided dimension $d$ satisfies $d\geq 3$. On the opposite, in dimension two, relaxing incompressibility leads to a moduli space of infinite dimension (see Propositions 4.11 and 4.12).

1.1. Organization of the paper

In Section 2, we provide background on affine connections.

In Section 3, we give the expression of symmetric affine connections in projective charts (where unparameterized geodesics coincide with the straight lines of the chart). We also give formulas to compute the curvature and Ricci tensors. In this class, incompressible connections (connections locally preserving a volume form) are determined by the logarithmic derivative of a potential $\psi$.

In Section 4, we give the classification of stiff connections in local charts. We define the canonical models and explore (non-)existence of isomorphisms between neighbourhoods of one or two models and their classification. We prove that incompressibility hypothesis is automatically satisfied for $d\geq 3$ (Proposition 4.10). On the opposite, for $d=2$, relaxing the hypothesis leaf to infinite dimensional families of connections.

In Section 5, we compare various flavours of conformality for connections. In particular, we prove in Proposition 5.3 that for a connection whose geodesics coincide with straight lines of some pseudo-Euclidean structure, local preservation of the conformal structure by the holonomy implies that the connection is flat.

In Section 6, we provide explicit computations for the parallel transport in canonical models $\mathcal{S}_{\lambda}^{\pm}(p,d-p)$.

In Section 7, we present the most remarkable feature of stiff connections. In the same conformal class, we obtain three pseudo-Riemannian metrics:

• •

the usual flat metric $g$ of the underlying pseudo-Euclidean space;

• •

the Ricci tensor (or its opposite) of connection $\nabla$ (always nondegenerate);

• •

an isochrone metric $h$ such that the geodesics of $\nabla$ have constant speed with respect to $h$ (but fail to be geodesics for the Levi-Civita connection defined by $h$).

These three metrics are related to each other by a potential $\psi$ involved in the computation of the coefficients of $\nabla$.

In Section 8, we investigate the global structure of pairs $(M,\nabla)$ formed by a (similarity) pseudo-Euclidean manifold $M$ and a stiff connection $\nabla$. In particular, we describe two families of remarkable examples:

• •

compact manifolds endowed with a stiff connection: their canonical model satisfies $\lambda=0$ and they are never geodesically complete;

• •

for each dimension $d\geq 3$, there is only one geodesically complete pair $(M,\nabla)$, and they only occur in signatures $(d,0)$ and $(0,d)$. It is the open ball with a connection reminiscent of both the Cayley-Klein rectilinear model but also of the conformal structure of the Poincaré hyperbolic model. In dimension $d=2$ we raise the question whether this is also the only one, and translate this into an analytic question about meromorphic functions on the unit disk.

Acknowledgements.

The authors would like to thank Charles Frances and François Labourie for valuable remarks about compact manifolds locally modelled on pseudo-Euclidean spaces (Remarks 8.10 and 8.11).

2. Reminder on affine connections

In this paper, we study affine connections on a smooth manifold $M$ (locally diffeomorphic to ${\mathbb{R}}^{d}$). We use the covariant derivative point of view. We assume that the coefficients of the connections are always smooth.

Notations. It is usual, and convenient, when dealing with tensors, to denote the $d$ coordinates of points and vectors in a dimension $d$ chart using exponents as in $x^{1}$, $x^{2}$, …, $x^{d}$. However we chose to use $x_{1}$, $x_{2}$, …, $x_{d}$ instead. The reason is that, starting from Section 4, we make abundant use of quadratic forms and we prefer to write $x_{1}^{2}-x_{2}^{2}$ instead of $(x^{1})^{2}-(x^{2})^{2}$, and want to maintain a coherent notation throughout the article. For all other types of tensors (forms, curvature, etc.) we use the traditional notations and conventions.

Affine connection. Denote $(e_{1},\dots,e_{d})$ the canonical basis of ${\mathbb{R}}^{d}$. Let $U$ be an open subset of ${\mathbb{R}}^{d}$ considered as a chart of a manifold, and let $X$, $Y$ be vector fields on $U$. We recall that $\partial_{X}Y$ denotes the directional derivative222Recall that the vector field $\partial_{X}Y$ depends on the chart and will not be transported in a compatible way by a change of variable $\phi$, i.e. most of the times $\phi_{*}(\partial_{X}Y)\neq\partial_{\phi_{*}X}\phi_{*}Y$. of $Y$ along $X$ in the chart $U$:

[TABLE]

In the chart, an affine connection $\nabla$ is expressed as follows: to vector fields $X$, $Y$ it associates

[TABLE]

where $\Gamma$ is a bilinear endomorphism of ${\mathbb{R}}^{d}=T_{x}U$ for each $x\in U$:

[TABLE]

where the coefficients $\Gamma^{i}_{jk}$ are $d^{3}$ functions of $x$ and are called the Christoffel symbols of the connection. The ordering of the two lower indices of $\Gamma^{i}_{jk}$ is not uniform among authors, so the formula above specifies our choice.

The canonical connection of ${\mathbb{R}}^{d}$ is defined by $\Gamma=0$, i.e.

[TABLE]

A connection on $M$ that is canonical in one chart will fail to be in many other charts. A connection for which there are local charts where it is canonical is called locally trivializable.

Parallel transport. For a curve $t\mapsto\gamma(t)$ whose differential does not vanish, the parallel transport of a vector $v$ along $\gamma$ with respect to the connection $\nabla$ is a vector $v(t)$ attached to $\gamma(t)$ and that is locally a solution of

[TABLE]

for any vector fields $X$, $Y$ such that $X(\gamma(t))=\dot{\gamma}(t)$ and $Y(\gamma(t))=v(t)$ and with given value at initial time $t_{0}$ (which is often $t_{0}=0$). This amounts to the ordinary differential equation

[TABLE]

which, omitting $t$, reads as

[TABLE]

This formula allows to generalize parallel transport to paths whose derivative vanishes for some values of $t$. The map that associates $v(t)$ to $v(0)$ can be seen as a map from the tangent space of $M$ at the point represented by $\gamma(0)$ to the tangent space of $M$ at the point represented by $\gamma(1)$. This map is also called parallel transport along $\gamma$.

Geodesics. The famous geodesic equation for a path $\gamma$ asks that $\dot{\gamma}(t)$ be a parallel transport along $\gamma$, i.e.

[TABLE]

It depends only on the quadratic vector form associated to the bilinear vector form $\Gamma$. It has the same geodesics as the symmetrized connection with $\Gamma^{\mathrm{sym}}(u,v)=\frac{1}{2}{}\left(\Gamma(u,v)+\Gamma(v,u)\right)$.

Symmetric connections (a.k.a. torsion free connections). The connection is symmetric whenever the bilinear form $\Gamma$ is symmetric, i.e.

[TABLE]

This is independent of the chart and can be defined in a coordinate-independent way by the cancellation of an associated tensor called the torsion, but we will not use this here.

In dimension $d=1$ all connections are symmetric.

Holonomy. The parallel transport along a closed curve is called a holonomy and is a self-map of the tangent space at $\gamma(0)$.

A connection is called flat333Some authors use the term parallelizable because this is in relation with invariant frame fields. This may be less ambiguous than flat in a situation where the connections are allowed to not be symmetric, or in situation where several notions of flatness occur (conformal flatness, projective flatness, etc.). if, locally, its holonomies are all the identity. By this we mean that every point has a neighbourhood $U$ such that the holonomy of every path contained in $U$ is the identity. This is equivalent to asking that the curvature tensor (see below) vanishes everywhere.

A connection is locally trivializable if and only if it is symmetric and flat (see [KN63], Chapter II, Section 9).

In dimension $d=1$ a connection is automatically locally trivializable.

Curvature tensor. To a connection $\nabla$, one associates its curvature tensor $R$. It is a $(1,3)$-tensor. There are two conventions for its definition, depending on the author(s), and they give opposite tensors. The convention we choose is, among the articles and books we cite, the one followed by [KN63, Kob08, NS94, Nur12, Sch54, Spi99, Sze04, Sch54] and we believe [Car24], but the opposite of [Bes87] (we do not know about [Tho25, Wol62]). To our knowledge, all these authors agree on the Ricci tensor444Authors using the other convention than us for $R$ systematically take $R(X,Z)$ to be the trace of $Y\mapsto R(X,Y)Z$ instead of taking $R(Y,Z)$ to be the trace of $X\mapsto R(X,Y)Z$, resulting in the two Ricci tensors being not opposite but equal. and on the scalar curvature deduced from the latter. We will not give the definition of $R$ here but we will use that its coefficients in charts are given by (see Proposition 7.6 page 145 in Chapter III of [KN63] for a reference555They decompose $R(X,Y)Z$ as $\sum R^{i}_{jkl}X_{k}Y_{l}Z_{j}e_{i}$. Hence what we denote $i,j,k,l$ they denote respectively $k,l,j,i$. Moreover in the coefficients of $R$ they put first the lower index that we put last.)

[TABLE]

and that it gives the first-order infinitesimal holonomy of the connection, as explained below. As for $\Gamma^{i}_{jk}$, the ordering of the three lower indices of the coefficients of $R$ varies among authors.

In dimension $d=1$ the curvature tensor is automatically [math].

Infinitesimal holonomy. For a symmetric affine connection $\nabla$ on an open subset of $\mathbb{R}^{d}$, we consider the holonomy of the parallel transport around a small rectangular flag based at some point, directed by vectors $e_{i}$ and $e_{j}$ and positively oriented with respect to the basis $(e_{i},e_{j})$. It is an endomorphism of the tangent space at the point and as the diameter of the rectangle tends to [math], it expands as ${\operatorname{Id}}+\mathcal{A}\mathcal{M}+o(\mathcal{A})$, where $\mathcal{A}$ is the area of the rectangular flag for the canonical Euclidean metric of ${\mathbb{R}}^{d}$.

The endomorphism $\mathcal{M}$ will be referred to here as the first order infinitesimal holonomy of the connection at this point, and we will omit the mention of “first order” in the sequel.

It depends on the pair $(i,j)$ and we denote by

[TABLE]

the corresponding matrix in the canonical basis of ${\mathbb{R}}^{d}$. The coefficient at the $l$-th line and $k$-th column of $M(i,j)$ is equal to $-R_{ijk}^{l}$: this is for instance proved in [Sch54], Chapter III, Section 4, Equation (4.11) page 140.

Riemannian metrics. Given a $C^{1}$ Riemannian metric $g$, there is a unique symmetric connection whose parallel transport preserves $g$. It is called the Levi-Civita connection of $g$. Its parallel transport maps, in particular its holonomies, are isometries between tangent spaces, with respect to $g$.

Change of variable. If $\phi$ is a diffeomorphism between open subsets of ${\mathbb{R}}^{d}$, and $\Gamma$ the Christoffel symbol of a connection in some chart then the image of this connection by $\phi$ has Christoffel symbols $\widetilde{\Gamma}$ that are expressed as follows:

[TABLE]

where the linear map $L\in M_{d}(\mathbb{R})$ is the differential of $\phi$, $H:{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ is the Hessian bilinear form of $\phi$, and $(L,L)$ is the linear map $(u,v)\mapsto(L(u),L(v))$.

If the change is just linear: $\phi=L$, then

[TABLE]

Pulling back the canonical connection. Coming back to the general case, equation 2.1 allows to give a simple expression of the pull-back $\nabla$ of the canonical connection $\widetilde{\nabla}$ (for which $\widetilde{\Gamma}=0$):

[TABLE]

As an application, we compute the pull-back $\nabla$ of the canonical connection by a projective transformation $\phi$ of ${\mathbb{R}}^{d}$.

We write $\phi(x)=\frac{1}{F(x)}M(x)$ where $M=M_{0}+M^{\prime}$, $M_{0}\in{\mathbb{R}}^{d}$, $M\in\mathrm{GL}({\mathbb{R}}^{d})$, $F=F_{0}+F^{\prime}$ where $F_{0}\in{\mathbb{R}}$ and $F^{\prime}$ is a linear form on ${\mathbb{R}}^{d}$. Then

[TABLE]

with

[TABLE]

i.e. $a=-dF/F$.

This will be generalized in Section 3.4.

3. Tensor formulas for projective charts

A special class of affine connections admits charts conjugating geodesics of the connection to straight lines of the chart. Their existence implies strong constraints on the coefficients of the connection. A symmetric connection having such charts near every point is sometimes called projectively flat, by reference to the projective space $\mathbb{RP}^{d}$ and its group of transformations. We will call here projective chart or projective flattening a chart where a (not necessarily symmetric) connection has all its geodesics that are straight lines of the chart.

In dimension one all connections are symmetric and it is obvious that all connections are not only projectively flat, but also that any chart is a projective flattening, making the theory almost pointless. However for technical reasons related to potential and Proposition 3.16, this helps to determine the parameterization of geodesics in any dimension, so we will also cover this case.

In this section and further, we make use of the Kronecker symbol $\delta_{ij}=0$ if $i\neq j$ and $1$ if $i=j$.

3.1. The associated form

In this section, we investigate the form that a connection takes in a projective chart. These results are not new. In fact Herman Weyl already proved a more general statement, see Proposition 17 page 251 in volume 2 of [Spi99].

We give a first characterization for a connection that is not necessarily symmetric.

Lemma 3.1**.**

Consider an open subset $U$ of $\mathbb{R}^{d}$ and a general connection $\nabla$ on $U$, with symbol $\Gamma$. The following statements are equivalent:

1. (1)

All geodesics are straight lines of $\mathbb{R}^{d}$. 2. (2)

Equations (3.1) and (3.2) hold:

[TABLE]

By specifying to $i=j$, equation (3.2) above reads $\Gamma_{ii}^{k}=0$ when $k\neq i$. Of course when $d=1$, all the conditions of (2) are empty.

Proof.

We remind that the geodesic equation reads $\gamma^{\prime\prime}(t)=-\Gamma(\gamma^{\prime}(t),\gamma^{\prime}(t))$. So the condition on geodesics is equivalent to

[TABLE]

We are now reduced to a simple linear algebra problem.

By applying this to $v=e_{i}$, we get that $\Gamma_{ii}^{k}=0$ for all $k\neq i$. By applying to $v=e_{i}+\lambda e_{j}$ when $i\neq j$, we get that

[TABLE]

must be colinear to $e_{i}+\lambda e_{j}$ (exponents on $\lambda$ means power, not index), hence

[TABLE]

for all $\lambda\in\mathbb{R}$ and for $k\notin\{{i,j\}}$ and

[TABLE]

for all $\lambda\in\mathbb{R}$. Then from looking at the $\lambda^{1}$ coefficient of these identities we get identities (3.1) and (3.2).

For the converse by substituting and grouping one eventually finds that if equations (3.1) and (3.2) are satisfied, we have

[TABLE]

∎

For symmetric connections, these constraints are strong enough to give an explicit and simple characterization. In particular, we give a general expression for the curvature tensor $R$ associated to $\nabla$.

Corollary 3.2**.**

Consider an open set $U\subset\mathbb{R}^{d}$ and a symmetric connection $\nabla$ of symbol $\Gamma$. The following statements are equivalent:

1. (1)

All geodesics are straight lines of $\mathbb{R}^{d}$. 2. (2)

There exists a $1$-form $a=\sum_{i}a_{i}dx_{i}$ such that

[TABLE]

In this case, for any $i,j,k$, the coefficients of $\Gamma$ satisfy

[TABLE]

and $a$ is unique. Besides, the coefficients of the curvature tensor $R$ associated to $\nabla$ are given by

[TABLE]

where $\delta_{ij}$ is the Kronecker symbol.

Proof.

Adding the symmetry condition to (3.2) we obtain that $\Gamma_{ij}^{k}=0$ for any choice of $i,j,k$ provided $k\notin\{{i,j\}}$. We also obtain that $\Gamma_{ii}^{i}=2\Gamma_{ij}^{j}$ for any choice of $i,j$ such that $i\neq j$. We set $a_{i}=\frac{1}{2}\Gamma_{ii}^{i}$. The general formula $\Gamma_{ij}^{k}=\delta_{ik}a_{j}+\delta_{jk}a_{i}$ and the expression of $\Gamma(u,v)$ given in the statement immediately follow.

Conversely a connection as in (2) satisfies the relations of (3.2) and since $\Gamma_{ii}^{i}=2a_{i}$, the $a_{i}$ are unique.

The coefficients of $R$ in coordinates are given by

[TABLE]

We obtain the following formula for $\Gamma_{ip}^{l}\Gamma_{jk}^{p}-\Gamma_{jp}^{l}\Gamma_{ik}^{p}$:

[TABLE]

Summing on every index $p$, the last two terms are cancelled out and we obtain:

[TABLE]

The final formula follows. ∎

In dimension $d=1$ the conditions above are always satisfied and the only coefficient of $a$ and the only coefficient of $\Gamma$ are related by $2a_{1}=\Gamma_{11}^{1}$. Besides, the curvature tensor is always [math].

Definition 3.3**.**

In the present article, the differential form

[TABLE]

appearing in the previous statement will be called the associated form.

Unfortunately, $a$ is not independent, as a $1$-form, of the projective chart in which we work, as the example at the end of Section 2 shows ($a=0$ in one chart and $a=-dF/F$ in the other), unless we restrict to ${\mathbb{R}}$-affine changes of charts, see Section 3.3.

Remark 3.4*.*

For a symmetric connection, if the curvature tensor uniformly vanishes, a classical theorem (see [Sze04], section “Locally flat spaces” on page 532) implies that it is locally trivializable (in the sense of Section 2, i.e. there is a chart in which it is the canonical connection). If the curvature of a connection vanishes uniformly on a given projective chart, any local trivializing change of charts is then an injective map locally preserving straight lines of $\mathbb{R}^{d}$, because it maps geodesics to geodesics. If $d\geq 2$, it is known (see [Löw82]) that such a map must be locally a projective transformation of $\mathbb{R}^{d}$. Hence in this situation, for every point there is a neighbourhood and a projective transformation defined on it that trivializes the connection.

Remark 3.5*.*

H. Cartan in [Car24] and independently T.Y. Thomas in [Tho25] proved that, given any connection and its expression in any chart, the local existence of projective charts is equivalent to the vanishing of a specific tensor, whose expressions we will not give here, as we shall not make use of this. The interested reader can for instance consult Theorem 1.7 in [Nur12].

A remarkable property of projective charts of symmetric connections is that parallel transport preserves planes.

Corollary 3.6**.**

Assume that $d\geq 2$ and $U\subset{\mathbb{R}}^{d}$ is a projective chart of a symmetric connection $\nabla$. If $\gamma$ is a path contained in $U$ and in an affine $2$-plane $P$ of ${\mathbb{R}}^{d}$ and $v$ a vector contained in the vector subspace $\vv P$ of ${\mathbb{R}}^{d}$ associated to $P$, then the parallel transport of $v$ along $\gamma$ still belongs to $\vv P$.

Proof.

From the formula $\Gamma(u,v)=a(u)v+a(v)u$ it follows that $\Gamma(u,v)$ belongs to the vector space generated by $u$ and $v$. Recall the parallel transport equation: $\dot{v}=\Gamma(\dot{\gamma},v)$ where $\Gamma$ is taken at point $\gamma(t)$. It follows that if $v$ and $\dot{\gamma}$ belong to $\vv P$ then $\dot{v}$ too. The restriction of the connection $\nabla$ to the subspace $P$ of ${\mathbb{R}}^{d}$ thus makes sense, and defines a parallel transport $v(t)$ of $v$ along $\gamma$, for which $v(t)\in\vv P$, and which satisfies the same equation $\dot{v}=\Gamma(\dot{\gamma},v)$. The result follows by uniqueness of solutions of differential equations. ∎

3.2. Ricci tensor and incompressible connections in projective charts

The Ricci tensor $\mathrm{Ric}$ is a $(0,2)$-tensor obtained by contraction of the curvature tensor $R$. Its coefficients are given (with our choices of conventions for $R$) by

[TABLE]

Thus, in a projective chart of a symmetric affine connection $\nabla$ on a $d$-dimensional smooth manifold, where $a_{1},\dots,a_{d}$ are the coefficients of the associated form $a$, the coefficients of the Ricci tensor deduced from equation (3.4) are

[TABLE]

Definition 3.7**.**

A connection is called incompressible whenever near every point there exists a volume form that is locally preserved by the parallel transport.

By [NS94], Chapter I, Proposition 3.1, this is equivalent to the first order infinitesimal holonomy having vanishing trace everywhere. It is easy to see that in dimension one a connection is necessarily incompressible (it is always locally trivializable).

Proposition 3.8**.**

Given a pair $(M,\nabla)$ formed by a $d$-dimensional smooth manifold and an affine symmetric and projectively flat connection, we consider a projective flattening $\phi$. Let $a_{1},\dots,a_{d}$ be the coefficients of $\nabla$ in chart $\phi$. The following statements are equivalent:

1. (1)

The associated form $a$ is closed. 2. (2)

There exists a locally defined nowhere vanishing smooth function $\psi$, called a potential, such that for any index $i$, $a_{i}=-\frac{\partial_{i}\psi}{\psi}$ (i.e. $a=\frac{-1}{\psi}d\psi$). It is unique up to multiplication by an element of ${\mathbb{R}}^{*}$. 3. (3)

The Ricci tensor of $\nabla$ is symmetric ($\mathrm{Ric}_{jk}=\mathrm{Ric}_{kj}$* for any pair of indices $j,k$).* 4. (4)

The connection $\nabla$ is incompressible.

*We define a connection satisfying these conditions as an incompressible projectively flat connection.

In these cases, the preserved volume forms are locally given by*

[TABLE]

for $\alpha\in{\mathbb{R}}$, and the coefficients of the Ricci tensor are given by

[TABLE]

Besides, the coefficients of the curvature tensor are completely determined by the Ricci tensor: if $d=1$ then $R=0$ and $\mathrm{Ric}=0$; if $d\geq 2$ then

[TABLE]

We can also express $R$ in terms of $\psi$:

[TABLE]

Proof.

By Equation (3.5) we have

[TABLE]

This proves that the Ricci tensor is symmetric if and only if the condition $\partial_{k}a_{j}=\partial_{j}a_{k}$ is satisfied for any pair of indices $j,k$. This is equivalent to saying that the differential form $a=\sum_{i}a_{i}dx_{i}$ is closed.

Closedness of $a$ is equivalent to the existence of a locally defined function $\theta$ such that $a=d\theta$. We define $\psi=e^{-\theta}$ and obtain that $a=d\theta=-\frac{d\psi}{\psi}$. Conversely if (2) holds then $a=d\theta$ for $\theta=-\log|\psi|$. We proved that statements $(1)$, $(2)$ and $(3)$ are equivalent.

The connection $\nabla$ locally preserves some volume form if and only if its first-order infinitesimal holonomy matrices are traceless (see [NS94], Chapter 1, Section 3, Proposition 3.1). This is equivalent to saying that for any pair of indices $i,j$, we have $\sum_{k}R_{ijk}^{k}=0$. By formula (3.4), we have

[TABLE]

Statement (4) is thus equivalent to closedness of differential form $a$.

Near a given point, invariant volume forms are locally determined by the choice of a volume form at this point, since one can transport this choice on a whole small neighbourhood by parallel transport and by incompressibility, this choice is coherent. The obtained volume form depends linearly on the initial choice, which occurs in a one-dimensional space, and any invariant form must locally be obtained that way. To prove that $\frac{dx_{1}\wedge\cdots\wedge dx_{d}}{\psi^{d+1}}$ is invariant, we could take this formula and differentiate it but we find interesting to show how this formula pops up as a consequence of analysing the parallel transport.

Consider an initial point $x_{*}\in{\mathbb{R}}^{d}$, choose one of the base vectors $e_{i}$, and define the path $\gamma(t)=x_{*}+te_{i}$. Let $v(0)$ be any vector and $v(t)$ be the parallel transport of $v$ along $\gamma$. By definition $\dot{v}=-\Gamma(\dot{\gamma},v)=\frac{1}{\psi}\left(d\psi(\dot{\gamma})v+d\psi(v)\dot{\gamma}\right)$ where $\Gamma$, $\psi$ and $d\psi$ are taken at the point $\gamma(t)$ and for any function $f$ of $t$, $\dot{f}$ means $f^{\prime}(t)$. For our particular choice of $\gamma$ this gives

[TABLE]

Note that $d\psi(e_{i})=\dot{\psi}=(\psi\circ\gamma)^{\prime}(t)$.

We choose $v(0)=e_{j}$. Then, as we already saw, $v(t)$ remains in the vector subspace generated by $e_{i}$ and $e_{j}$.

If $j\neq i$ then let us write $v(t)=f(t)e_{i}+g(t)e_{j}$ with $f(0)=0$ and $g(0)=1$. Then the differential equation reads $\dot{f}e_{i}+\dot{g}e_{j}=(\cdots)e_{i}+\frac{\dot{\psi}}{\psi}ge_{j}$. It follows that $\dot{g}/g=\dot{\psi}/\psi$, hence $g(t)=\psi\circ\gamma(t)/\psi(x_{*})$.

If $j=i$ then let us write $v(t)=f(t)e_{i}$ with $f(0)=1$ and the differential equation reads $\dot{f}e_{i}=2\frac{d\psi(e_{i})}{\psi}fe_{i}=\frac{2\dot{\psi}}{\psi}fe_{i}$. Hence $\dot{f}/f=2\dot{\psi}/\psi$, so $f(t)=(\psi\circ\gamma(t)/\psi(x_{*}))^{2}$.

It follows that the form $\frac{dx_{1}\wedge\cdots\wedge dx_{d}}{\psi^{d+1}}$ is invariant by the parallel transport along lines parallel to $e_{i}$. And this holds for any $i$. Any volume form which is invariant by parallel transport must be invariant by transport along these lines, so must be proportional to the above form. Since we know by incompressibility that there is an invariant form, the above form is invariant.

The last expressions for $\mathrm{Ric}$ and $R$ are obtained by simple computations, using for $\mathrm{Ric}$ equation (3.5) and point (2) of the present proposition, and for $R$ equations (3.4) and (3.5) and the fact that $a$ is closed. ∎

Remark 3.9*.*

Be warned that the potential is non-unique and a priori only defined locally. If $U\subset{\mathbb{R}}^{d}$ is not simply connected, it could well happen that $\psi$ cannot be globally defined on $U$, in the sense that there be monodromy factors to take into account. For instance, let $U={\mathbb{R}}^{2}-\{{O\}}$, where $O$ is the origin, and $\psi=\exp(\theta)$ where $\theta$ is the polar angle of vector $(x,y)$. Here the monodromy factor is $\exp(2\pi)$.

Remark 3.10*.*

Note also that the potential behaves like a function under an affine change of variable, but not under a projective change of variable. See Section 3.3.

Below we prove that a projectively flat connection is flat if and only if it is Ricci flat.

Proposition 3.11**.**

Given a pair $(M,\nabla)$ formed by a smooth manifold and an affine symmetric connection,666Note that we do not assume incompressibility a priori. we consider a projective chart $\phi:U\to{\mathbb{R}}^{d}$. The following statements are equivalent:

1. (1)

$\nabla$* is Ricci flat (i.e. $\mathrm{Ric}=0$) on $U$;* 2. (2)

$\nabla$* is flat (i.e. $R=0$) on $U$;* 3. (3)

$d=1$* or near every point of $\phi(U)$ one can find a projective change of chart that trivializes $\nabla$.*

Proof.

If $d=1$ then $R=0$ and $\mathrm{Ric}=0$ so it is enough to treat the case $d\geq 2$. Since $R$ and $\mathrm{Ric}$ are tensors, if they vanish in a chart then they vanish for any other chart $\tilde{\phi}$ on the corresponding points, if any, i.e. on $\tilde{\phi}(U)$.

We first assume that $\mathrm{Ric}$ vanishes. In particular, we have $\partial_{i}a_{j}=\partial_{j}a_{i}$ for any pair of indices $i,j$. By equation (3.5), the condition $\mathrm{Ric}_{ij}=0$ then simplifies to $a_{i}a_{j}=\partial_{i}a_{j}$. Injecting these identities into equation (3.4), we obtain that every coefficient of the curvature tensor $R$ vanishes.

If $R=0$ then the connection $\nabla$ is trivial up to a projective transformation by Remark 3.4. We can also prove this directly by solving the equations $\partial_{i}\partial_{j}\psi=0$: they imply that $\psi=c+\sum b_{i}x_{i}$ for some constants $c$ and $b_{i}$. If $\psi=c$ then $a=0$ and $\nabla$ is already trivial. Otherwise according to equation (2.3), $\nabla$ coincides with the pull-back of the canonical connection by any projective transformation of denominator $\psi$. So $\phi$ trivializes $\nabla$.

Of course, the canonical connection has vanishing curvature and Ricci tensor. ∎

3.3. Affine change of variable

Here we explore the effect of a linear change of variable on the coefficients $a_{1},\dots,a_{d}$ of the associated form of a projective chart of a symmetric connection.

Proposition 3.12**.**

Consider a manifold $M$ with a symmetric connection having a projective chart $\phi:O\subset M\to U\subset{\mathbb{R}}^{d}$ with associated form $a=\sum_{i}a_{i}dx_{i}$. If $A$ is an affine change of variable then $A\circ\phi$ is also a projective chart. Denote $L$ the linear part of $A$ and $\tilde{a}=\sum_{i}\tilde{a}_{i}dx_{i}$ the associated form for this new chart. We have:

[TABLE]

This notation is a shorthand for: $\forall x\in U$, for all $u\in\mathbb{R}^{d}$, $\tilde{a}\{A(x)\}(u)=a\{x\}(L^{-1}(u))$ where $a\{x\}(u)=\sum_{i}a_{i}(x)u_{i}$ and $\tilde{a}\{A(x)\}(u)=\sum_{i}\tilde{a}_{i}(A(x))u_{i}$.

Proof.

This is an easy consequence of equations (2.2) and (3.3). Indeed, consider the first one: $\tilde{\Gamma}\{A\}\circ(L,L)=L\circ\Gamma$. Substitute the second one on each side:

[TABLE]

By composing on the right with $(L^{-1},L^{-1})$ we get that

[TABLE]

for all $u,v$. By uniqueness of the $a_{i}$ in Corollary 3.2, it follows that $\tilde{a}\{A\}=a\circ L^{-1}$. ∎

Remark 3.13*.*

Note that (3.6) is exactly the same formula as for differential forms. In other words, $a$ behaves as a $1$-form provided we only perform affine changes of variables. This is the case of the change of charts of (similarity) pseudo-Euclidean manifolds.

Corollary 3.14**.**

Under the same assumptions, if $\nabla$ is incompressible, then the potential $\tilde{\psi}$ in the new chart and the potential $\psi$ in the original chart are related by:

[TABLE]

Thanks to this, one can speak of the potential of a connection defined on a finite-dimensional affine space $E$, or an open subset thereof, extending the case $E={\mathbb{R}}^{d}$.

3.4. Projective change of variable

Under a projective change of variable, lines are mapped to lines and a projectively flattened connection will remain projectively flattented. In this section we compute the action on the associated “form” $a$, and if the connection is incompressible, on the associated potentials $\psi$.

Proposition 3.15**.**

Let $\nabla$ a projectively flattened connection on an open subset of ${\mathbb{R}}^{d}$, with associated form $a$. Let $P(x)=\frac{1}{C(x)}F(x)$ be a projective transformation, with $F:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ and $C:{\mathbb{R}}^{d}\to{\mathbb{R}}$ affine maps. Denote $\tilde{\nabla}=P_{*}\nabla$ and $\tilde{a}$ the $1$-form associated to $\tilde{\nabla}$. Then:

[TABLE]

If moreover $\nabla$ is incompressible, let $\psi$ be a potential (defined locally or globally). Then the function $\tilde{\psi}$ defined by

[TABLE]

is a potential for $\tilde{\nabla}$.

Proof.

By Equation (2.1), we have

[TABLE]

where $\Gamma$ is taken at $x$, $L$ is the differential of $P$ at $x$, $H$ is the Hessian differential of $P$ at $x$ and $\tilde{\Gamma}$ is taken at $y=P(x)$. By definition, $\tilde{\Gamma}(u,v)=\tilde{a}(u)v+\tilde{a}(v)u$. Since $\tilde{a}\circ L=P^{*}(\tilde{a})$ where $\tilde{a}$ is taken at $y=P(x)$ and $P^{*}(\tilde{a})$ at $x$, we get

[TABLE]

Let us compute the second term of the subtraction. One finds that $L(u)=\frac{dF(u)}{C}-\frac{dC(u)}{C^{2}}F$ and

[TABLE]

where $F$, $C$ and all differentials $d(\cdots)$ are taken at $x$. So

[TABLE]

hence $-L^{-1}H(u,v)=\frac{dC(u)}{C}v+\frac{dC(v)}{C}u$. It follows that the forms $f=P^{*}\tilde{a}$ and $f=a+\frac{dC}{C}$ generate the same vector valued bilinear form $(u,v)\mapsto f(u)v+f(v)u$ and thus are equal as can be seen by taking $u=v=e_{i}$.

The claim on the potential easily follows by differentiation, recalling that $a=-d\psi/\psi$. ∎

3.5. Restriction to an affine subspace

Consider a projectively flattened connection $\nabla$ defined on an open subset $U$ of an affine space $E$. Denote $a$ its associated form. We saw in Corollary 3.6 that in the case $E={\mathbb{R}}^{d}$, the parallel transport along an affine $2$-plane $P$ of ${\mathbb{R}}^{d}$ preserves vectors tangent to $P$. This generalizes to any affine subspace.

Proposition 3.16**.**

For any affine subspace $F$ of $E$:

• •

for any point $x\in E$, $\Gamma\{x\}(\vec{F},\vec{F})\subset\vec{F}$;

• •

$\nabla$* has a well-defined restriction $\nabla^{F}$ to $F$, given by the restriction of $\Gamma$;*

• •

the restriction of $a$ to $F$ is the associated form of $\nabla^{F}$;

• •

parallel transport of a vector in $\vec{F}$ along a curve contained in $F$ remains in $\vec{F}$;

• •

$\nabla^{F}$* is projectively flattened.*

Assume moreover that $\nabla$ is incompressible. Then

• •

$\nabla^{F}$* is incompressible;*

• •

for any potential $\psi$ (see Proposition 3.8) defined on an open subset $V$ of $U$, the restriction of $\psi$ to $F$ is a potential for $\nabla^{F}$ on $F\cap V$.

Proof.

By Equation 3.3, $\Gamma(u,v)=a(u)v+a(v)u$. The first three points immediately follow. The fourth point is a consequence of uniqueness of solutions of differential equation. The connections $\nabla$ and $\nabla^{F}$ having the same geodesics in $F$, the fifth point is immediate. If $\nabla$ is incompressible then $a$ is closed, so the restriction of $a$ to $F$ is closed, so it is incompressible (see Proposition 3.8). The relation $a=-d\psi/\psi$ still holds for the restrictions of $a$ and $\psi$ to $F$, so by Proposition 3.8, the seventh point follows. ∎

3.6. Geodesics

Since any geodesic stays in a straight line $L\subset E$, the geodesic equation $\ddot{\gamma}=-\Gamma(\dot{\gamma},\dot{\gamma})$ is easy to solve. Note that since $L$ is one-dimensional, the restriction of $\nabla$ to $L$ is necessarily incompressible, and thus has a potential. In the case the original connection is incompressible, the restriction of a potential to $L$ can be taken as a potential by Proposition 3.16. In any case, one can take $\psi=\exp(-\int a)$ where $\int a$ is any antiderivative of $a$ on $L$.

Now we have $\Gamma(\dot{\gamma},\dot{\gamma})=2a(\dot{\gamma})\dot{\gamma}=\frac{-2}{\psi}d\psi(\dot{\gamma})\dot{\gamma}$ so the geodesic equation reads

[TABLE]

which is equivalent to $\frac{d}{dt}(\dot{\gamma}/\psi(\gamma)^{2})=0$, i.e.

[TABLE]

If one chooses an affine parameterization $\phi:s\in{\mathbb{R}}\mapsto x\in L$, and denote $\psi_{s}=\psi\circ\phi$ (the index $s$ is purely symbolic) then we get the equation $\dot{s}/\psi_{s}^{2}(s)=\operatorname{cst}$. This is equivalent to

[TABLE]

for some $\alpha\in{\mathbb{R}}^{*}$ and $\beta\in{\mathbb{R}}$, where $F$ is an antiderivative of $1/\psi_{s}^{2}$. This can be rephrased as

[TABLE]

or as follows: the change of variable $F\circ\phi$ with $F^{\prime}=1/\psi_{s}^{2}$ trivializes the restriction of $\nabla$ to $L$.

This has the following interesting consequence: for any Euclidean or pseudo-Euclidean form $q$ on $\vec{E}$, every geodesic has constant speed with respect to the Riemannian or pseudo-Riemannian metric $g$ such that $g(dx,dx)=q(dx)/\psi(x)^{4}$ and we say that $\nabla$ and $g$ are isochrone to each other. It may seem somewhat arbitrary to have to choose some $q$ on $\vec{E}$ but in the context of stiff connections, there is (at least locally) a $q$ given a priori. We develop this in Section 7.1.

4. Classification of stiff connections

In this section we work directly in a pseudo-Euclidean affine space $E$ of dimension $d$ and signature $(p,d-p)$ and with a connection $\nabla$ defined on an open subset of $E$. This subset can for instance be a chart of a pseudo-Euclidean manifold $M$ or a similarity manifold. We only consider the case $d\geq 2$.

Note: Our convention in denoting the signatures is that $p$ denotes the number of $+$, $d-p$ the number of $-$, and there is not [math].

Given a pseudo-Euclidean vector space with associated bilinear form $q$, by an isometry we mean a linear self map $f$ the space such that $q\circ f=q$. In particular we exclude map for which we would have $q\circ f=-q$, like $(x_{1},x_{2})\mapsto(-x_{2},x_{1})$ in ${\mathbb{R}}^{2}$ endowed with $q(x)=x_{1}^{2}-x_{2}^{2}$. By a similarity, we mean the composition of a homothety and of an isometry. Note that a similarity multiplies the underlying quadratic form $q$ by a positive constant. Of course if we consider an affine space instead of a vector space, we add a translation term to isometries and similarities.

Recall that a connection $\nabla$ on the pseudo-Euclidean affine space $E$ is called stiff if:

• •

$\nabla$ is symmetric,

• •

geodesics of $\nabla$ are straight lines (not necessarily with constant speed vector),

• •

first order infinitesimal holonomies are isometries of $E$.

We will see in Section 4.2 that non-flat stiff connections are quite rigid in that they come, up to an affine isometry of $E$, in a one parameter family for each dimension and signature. And up to a similarity of $E$ they come up in a finite family with only three elements.

We will study a slightly relaxed condition where we only ask infinitesimal holonomies to be conformal for $E$, i.e. similarities. This will be called here weakly stiff connections. We will see that in dimension $d\geq 3$, weakly stiff connections are necessarily stiff, but that in dimension $2$ there form a wider class: actually they come in families of infinite dimension.

4.1. Infinitesimal holonomy and coefficients of the curvature tensor

The condition that the first-order infinitesimal holonomy of $\nabla$ consists only of infinitesimal similarities of $E$ can be directly translated into equations satisfied by the coefficients of the curvature tensor (and this is independent of the chart being projective or not). Let $E^{p,d-p}$ denote ${\mathbb{R}}^{d}$ equipped with the quadratic form $q(x)=x_{1}^{2}+\cdots+x_{p}^{2}-x_{p+1}^{2}-\cdots-x_{d}^{2}=\sum_{i}\epsilon_{i}x_{i}^{2}$, $\epsilon_{i}=\pm 1$. We denote $g$ the associated flat pseudo-Riemannian symmetric $(0,2)$-tensor, for which $g_{ii}=\epsilon_{i}$ and $i\neq j\implies g_{ij}=0$. It may also be denoted $g=\sum_{i}\epsilon_{i}\,dx_{i}\otimes dx_{i}$.

Lemma 4.1**.**

Given a connection $\nabla$, not necessarily symmetric, defined on an open subset of $E^{p,q-p}$, the first-order holonomy of $\nabla$ consists in infinitesimal similarities if and only if the coefficients of its curvature tensor $R$ satisfy the following conditions:

1. (1)

for any indices $i,j,k,l$ such that $k\neq l$, we have $R_{ijk}^{l}=-\epsilon_{l}\epsilon_{k}R_{ijl}^{k}$ (where $\epsilon_{1},\dots,\epsilon_{p}=1$ while $\epsilon_{p+1},\dots,\epsilon_{d}=-1$); 2. (2)

for any indices $i,j,k$, $R_{ijk}^{k}$ does not depend on $k$.

The first-order holonomy of $\nabla$ consists in infinitesimal isometries if and only if we have (1) and that for any indices $i,j,k$, $R_{ijk}^{k}=0$.

Proof.

In the canonical basis of $\mathbb{R}^{d}$, we denoted $M(i,j)$ the holonomy matrix for an infinitesimal rectangle based at some point and directed by vectors the $e_{i}$ and $e_{j}$. The matrix $M(i,j)$ has its coefficient at the $l$-th line and $k$-th column equal to $-R_{ijk}^{l}$.

The matrices $M(i,j)$ are infinitesimal similarities of the pseudo-Euclidean structure of the chart if and only if they are sums of a scalar multiple of the identity matrix and of a matrix $A$ such that $I_{p,d-p}A$ is antisymmetric ($I_{p,d-p}$ is the diagonal matrix whose $p$ first coefficients are equal to $1$ while the remaining $d-p$ are equal to $-1$). The condition on the coefficients of the curvature tensor follows.

Infinitesimal isometries correspond to the fact that the diagonal term vanishes. ∎

Recall that in dimension two, the curvature tensor is entirely determined by the four coefficients $R_{12k}^{l}$, since $R_{iik}^{l}=0$ and $R_{21k}^{l}=-R_{12k}^{l}$. It is well-known that the space $E^{1,1}$ has another coordinate system parameterized by $y\in{\mathbb{R}}^{2}$ in which the quadratic form takes the expression $y_{1}y_{2}$. In this coordinate system $E^{\prime}$, the above lemma becomes:

Lemma 4.2**.**

For $d=2$, on $E^{\prime}$, the first-order holonomy of $\nabla$ consists in infinitesimal similarities if and only if the coefficients of its curvature tensor $R$ satisfy $R_{122}^{1}=0$ and $R_{121}^{2}=0$. The first-order holonomy of $\nabla$ consists in infinitesimal isometries if and only if $R_{121}^{1}+R_{122}^{2}=0$.

4.2. Stiff connections

A weakly stiff connection on an open subset of $E$ is in particular projectively flattenend, i.e. all its geodesics are straight lines of $E$. A stiff connection is also incompressible. By Proposition 3.8, the coefficients of a projectively flattened, symmetric and incompressible connection $\nabla$ depend only on a potential function $\psi$. In the lemma below, we translate the conformality condition of infinitesimal holonomy into a system of partial differential equations satisfied by the potential.

Lemma 4.3**.**

Consider a projectively flattened, symmetric and incompressible connection $\nabla$ on $E$, with associated potential $\psi$. Then $\nabla$ is stiff if and only if:

• •

$\partial_{i}\partial_{j}\psi=0$* for any indices $i,j$ such that $i\neq j$;*

• •

there exists a smooth function $f$ such that $\partial_{i}^{2}\psi=\epsilon_{i}f$ for any index $i$ ($\epsilon_{i}=1$* if $i\leq p$ and $\epsilon_{i}=-1$ otherwise).*

Proof.

By Proposition 3.8,

[TABLE]

Following Lemma 4.1, using the hypothesis that $\nabla$ is incompressible, we get that the chart is a projective flattening if and only if the following condition holds:

1. (1)

for any indices $i,j,k,l$ such that $k\neq l$, we have $\epsilon_{k}R_{ijk}^{l}+\epsilon_{l}R_{ijl}^{k}=0$ (where $\epsilon_{1},\dots,\epsilon_{p}=1$ while $\epsilon_{p+1},\dots,\epsilon_{d}=-1$); 2. (2)

for any indices $i,j,k$, $R_{ijk}^{k}=0$.

The second point holds if and only if for any indices $i,j,k$, we have $\delta_{jk}\partial_{i}\partial_{k}\psi=\delta_{ik}\partial_{j}\partial_{k}\psi$. It follows then that for any $i\neq j$, we have $\partial_{i}\partial_{j}\psi=0$ (take $k=j$). Conversely if for all $i\neq j$, $\partial_{i}\partial_{j}\psi=0$ then by examining separately 4 case according to the values of the pair $(\delta_{ik},\delta_{jk})$, one sees that $\delta_{jk}\partial_{i}\partial_{k}\psi=\delta_{ik}\partial_{j}\partial_{k}\psi$ holds for all $i$, $j$ and $k$, hence (2) holds.

Now apply the first equation to $k=j$ and $l=i$ for $i\neq j$. We get $\epsilon_{j}R_{ijj}^{i}+\epsilon_{i}R_{iji}^{j}=0$. Using the expression recalled at the beginning of the proof we get $\epsilon_{i}\partial_{i}^{2}\psi=\epsilon_{j}\partial_{j}^{2}\psi$ for any pair of indices $i,j$. Conversely, if there is a smooth $f$ such that for all $i$, $\epsilon_{i}\partial_{i}^{2}\psi=f$ then $R_{ijk}^{l}=(\delta_{il}\delta_{jk}\epsilon_{j}-\delta_{ik}\delta_{jl}\epsilon_{i})\frac{f}{\psi}$ and thus $\epsilon_{k}R_{ijk}^{l}+\epsilon_{l}R_{ijl}^{k}=(\delta_{il}\delta_{jk}\epsilon_{j}\epsilon_{k}-\delta_{ik}\delta_{jl}\epsilon_{i}\epsilon_{k}+\delta_{ik}\delta_{jl}\epsilon_{j}\epsilon_{l}-\delta_{il}\delta_{jk}\epsilon_{i}\epsilon_{l})\frac{f}{\psi}$. Noticing that $\delta_{jk}\epsilon_{j}\epsilon_{k}=\delta_{jk}$ and using similar identities we get that $\epsilon_{k}R_{ijk}^{l}+\epsilon_{l}R_{ijl}^{k}=0$. ∎

Now, we are able to completely characterize potentials of stiff connections in pseudo-Euclidean spaces.

Theorem 4.4**.**

Consider a projectively flattened, symmetric and incompressible connection $\nabla$ defined on a connected open subset $U$ of $E^{p,d-p}$. Then $\nabla$ is stiff if and only if it has potential

[TABLE]

*i.e. $\psi=Kq/2+\Lambda$, where $K$ is a real constant, possibly null, and $\Lambda$ is a real affine function.

Conversely, any such potential $\psi$ that does not vanish on an open subset $U$ of $E$ defines a stiff connection on $U$.*

Proof.

Following Lemma 4.3, $\psi$, a priori only locally defined, satisfies a system of partial differential equations implying in particular that $\partial_{i}\psi$ depends only on $x_{i}$. Thus, $\epsilon_{i}\partial_{i}\partial_{i}\psi$ depends only on $x_{i}$. Since these functions do not depend on $i$, they are all equal to some constant $K$. We deduce that there are constants $K_{1},\dots,K_{d}$ such that $\partial_{i}\psi=K\epsilon_{i}x_{i}+K_{i}$. Finally, we obtain that, up to addition of a constant, $\psi$ is locally given by $\frac{K}{2}\sum_{i}\epsilon_{i}x_{i}^{2}+\sum_{i}K_{i}x_{i}$.

By connectedness of $U$ and uniqueness of the coefficients of a polynomial function on any small open subset of ${\mathbb{R}}^{d}$, the same $\psi$ can in fact be taken on all $U$ (caution must be taken because $\psi$ is only unique up to multiplication by a constant).

For the converse, any such $\psi$ satisfies the equations of Lemma 4.3 and thus define a stiff connection on $U\subset E$. ∎

As an immediate application of the formula expressing $\mathrm{Ric}_{ij}=(d-1)\frac{\partial_{i}\partial_{j}\psi}{\psi}$ in Proposition 3.8:

Corollary 4.5**.**

In the situation of Theorem 4.4, the Ricci tensor associated to $\nabla$ is proportional to the pseudo-Riemannian metric $g$ endowing $E$:

[TABLE]

where $K$ is the same as in Theorem 4.4.

Remark 4.6*.*

It follows that that $\nabla$ is flat (i.e. trivializable) if and only if $K=0$. In the latter case, flattenings are given by particular projective transformations (of course $\nabla$ may be already flattened: this is the case if and only if $\psi$ is constant). Indeed if $\nabla$ is flat then $R=0$, hence $\mathrm{Ric}=0$. Conversely if $\mathrm{Ric}=0$ then Proposition 3.11 proves that $\nabla$ is flat on the domain of $\phi$. Any flattening must preserve straight lines and for $d\geq 2$ such maps are known to be projective ([Löw82]).

Note that, though the potential $\psi$ is only uniquely defined up to multiplication by a constant, the coefficient $K/\psi$ is independent of this choice, and we have

[TABLE]

if $K\neq 0$, where $\tilde{\Lambda}(x)=\frac{2}{K}\Lambda(x)$, and of course

[TABLE]

if $K=0$.

Concerning the curvature tensor, Proposition 3.8 gives $R_{ijk}^{l}=(\delta_{il}\delta_{jk}-\delta_{jl}\delta_{ik})\epsilon_{k}\frac{K}{\psi}$. Hence:

Corollary 4.7**.**

In the situation of Theorem 4.4, for $i\neq j$ we have

[TABLE]

All other coefficients of $R$ vanish.

In other words, for $i\neq j$, the infinitesimal holonomy endomorphism (see Section 2) associated to $(e_{i},e_{j})$, has matrix $M(i,j)=\frac{K}{\psi}(\epsilon_{i}E_{ji}-\epsilon_{j}E_{ij})$ where $E_{ij}$ is the elementary matrix whose only non-zero entry is a $1$ at line $i$ and column $j$. This endomorphism also expresses as

[TABLE]

where $\cal J(i,j)$ is defined below.

Definition 4.8**.**

We let $\cal J(i,j)=\epsilon_{i}\,e_{j}\otimes e_{i}^{*}-\epsilon_{j}\,e_{i}\otimes e_{j}^{*}$. It is an infinitesimal isometry on the plane $\cal P=\operatorname{Vect}(e_{i},e_{j})$ and is [math] on $\cal P^{\perp}$. The restriction of $\cal J(i,j)$ to $\cal P$ has the following matrix in basis $(e_{i},e_{j})$

[TABLE]

Note that the coefficient $K/\psi$ can be interpreted as an angular speed for a (generalized) rotation.

Scalar curvature. For a pseudo-Riemannian metric, one of several equivalent definitions of the scalar curvature is as the trace of the $(1,1)$-tensor obtained from the $(0,2)$-tensor $\mathrm{Ric}$ via the musical isomorphism between $TM$ and $T^{*}M$. One can raise either the first or second variable: even though the two different operators thus obtained may differ, they have the same trace. For an arbitrary (symmetric) connection, there is no notion of scalar curvature, because there is no canonical choice of a musical isomorphism.

In our particular situation of a stiff connection, though, we can choose the pseudo-Euclidean metric $g$ to define the isomorphism. With this convention, we get what we call the scalar curvature $S(\nabla,g)$ of $\nabla$ with respect to $g$. From Corollary 4.5 the two endomorphisms whose trace can be taken are both equal to $(d-1)\frac{K}{\psi}\operatorname{Id}$ (they are equal because $\mathrm{Ric}$ is symmetric in our case), so

[TABLE]

Remark 4.9*.*

One may consider that $\nabla$ is positively curved at $x$ if $S(\nabla,g)$ is positive at $x$ and negatively curved if it is negative. The sign $S(\nabla,g)$ is the sign of $K/\psi$, i.e. the sign of $q(x)+\tilde{\Lambda}(x)$, and is thus locally constant (recall that the potential $\psi$ cannot vanish on the domain of definition of $\nabla$).

The connection preserves any affine $2$-plane $P$ of $E$, and if the restriction $\nabla^{P}$ of $q$ to $\overrightarrow{P}$ is nondegenerate, then the restriction of $\nabla$ to $P$ satisfies $S(\nabla^{P},g)=2K/\psi$, so has the same sign. In the case of a general (pseudo-)Riemannian metric, in dimension two, the classical scalar curvature is $d=2$ times the Gaussian curvature. From this, it makes sense to consider $K/\psi$ as the Gaussian curvature of $\nabla$ relative to $g$ and denote it $G(\nabla,g)$.

Interpreting the sign of the scalar curvature in dimension 2. In the case of signature $(2,0)$, we can informally state that, as in the Riemannian case, the holonomy of a small injective loop is close to be a rotation of a small angle and that turns in the same direction as the loop if $S(\nabla,g)>0$ and in the opposite direction if $S(\nabla,g)<0$. This can be generalized to a statement cover all signatures $(2,0)$, $(1,1)$ and $(0,2)$ as follows: the holonomy of a small injective loop is close to be an isometry close to the identity, which moves a vector $v$ not in the light cone by a small vector $dv$, such that the orientation of the basis $(v,dv)$ and of the loop are identical if $q(v)S(\nabla,g)>0$ and opposite if $q(v)S(\nabla,g)<0$. This last condition can be stated informally as: it pushes $v$ in the direction of the loop in the first case, and in the opposite direction in the second case.

Scalar curvature on nondegenerate subspaces. Consider any affine subspace $F$ of $E^{p,d-p}$ of dimension $d^{\prime}\geq 2$ such that the restriction of $q$ to $\vec{F}$ is nondegenerate and $F$ intersects the domain $U$ of a canonical model $\cal S$. Then by Proposition 3.16, $\nabla$ preserves $F$ and the restriction $\psi|_{F}$ of $\psi$ to $F$ is a potential for the restriction $\nabla^{F}$ of $\nabla$ to $F$. Hence $S(\nabla^{F},g^{F})=d^{\prime}(d^{\prime}-1)\frac{K}{\psi|_{F}}$. In particular,

[TABLE]

A remarkable consequence is that, at any given point of $F\cap U$, the scalar curvature of $\nabla^{F}$ has the same sign as the scalar curvature of $\nabla$.

4.3. Rigidity in dimension $d\geq 3$

It is well known that conformal geometry is much more rigid in dimension at least three than in dimension two. This phenomenon also appears in the study of weakly stiff connections, as will be shown by Propositions 4.10, 4.11 and 4.12.

Proposition 4.10**.**

If $d\geq 3$ then any weakly stiff connection is incompressible, i.e. stiff.

Proof.

Let us recall equation (3.4) for the expression of the curvature tensor $R$ in a pseudo-Euclidean chart:

[TABLE]

Following Lemma 4.1, the coefficients $R_{ijk}^{k}$ do not depend on $k$.

For any $i,j$ distinct, consider $k$ distinct from them (hypothesis $d\geq 3$ is crucial here). Substituting equation (3.4) in $R_{ijj}^{j}=R_{ijk}^{k}$, we deduce $a_{i}a_{j}-\partial_{i}a_{j}=0$ and thus $\partial_{i}a_{j}=\partial_{j}a_{i}$.

Proposition 3.8 then shows that the connection $\nabla$ is incompressible. ∎

Theorem 4.4 provides in particular a complete classification of weakly stiff connections in dimension higher than 2.

4.4. Flexibility in dimension two

For weakly stiff connections of signature $(1,1)$ or $(2,0)$, we will see that we have infinitely many degrees of freedom.

4.4.1. Signature $(1,1)$

Recall that the signature $(1,1)$ pseudo-Euclidean structure is given by the quadratic form $x_{1}^{2}-x_{2}^{2}$, which is isomorphic by a simple linear change of coordinates y=\left({\scriptstyle\begin{array}[]{rr}1&1\\ 1&-1\end{array}}\right)x to the quadratic form $y_{1}y_{2}$. We call $E^{\prime}$ the space ${\mathbb{R}}^{2}$ with coordinates $(y_{1},y_{2})$ and endowed with the quadratic form $y_{1}y_{2}$.

In the next statement we make use of $\hat{\mathbb{R}}={\mathbb{R}}\cup\{\infty\}$, which is a compact one dimensional manifold with charts $x\in\hat{\mathbb{R}}-\{\infty\}\mapsto x\in{\mathbb{R}}$ and $x\in\hat{\mathbb{R}}-\{0\}\mapsto 1/x\in{\mathbb{R}}$.

Proposition 4.11**.**

We consider a connection $\nabla$ defined on an open subset of $E^{\prime}$. We assume that $\nabla$ is symmetric and projectively flattened and denote $b=b_{1}dy_{1}+b_{2}dy_{2}$ the associated form (see Section 3.1 and Definition 3.3). Then $\nabla$ is weakly stiff if and only if $b$ is given by:

1. (1)

$b_{1}=\frac{1}{f_{1}(y_{2})-y_{1}}$* for some smooth function $f_{1}:U\to\hat{\mathbb{R}}$;* 2. (2)

$b_{2}=\frac{1}{f_{2}(y_{1})-y_{2}}$* for some smooth function $f_{2}:U\to\hat{\mathbb{R}}$.*

Conversely, any pair of smooth functions $f_{1},f_{2}$ defined on an open subset $U$ of $E^{\prime}$, mapping to $\hat{\mathbb{R}}$ and such that $f_{1}(y_{2})-y_{1}$ and $f_{2}(y_{1})-y_{2}$ do not vanish defines via the above formulas the associated form $b$ of a projectively flattened and weakly stiff connection.

Proof.

By Lemma 4.2, $\nabla$ is weakly stiff and only if $R_{121}^{2}=0$ and $R_{122}^{1}=0$ and by equation (3.4), this is equivalent to

• •

$\partial_{1}b_{1}=b_{1}^{2}$

• •

$\partial_{2}b_{2}=b_{2}^{2}$

so if $b_{1}$ vanishes at some $y_{*}$ then it must vanish on the whole connected component of $U\cap L$ where $L$ is the horizontal line trough $y_{*}$, and elsewhere $\partial_{1}(1/b_{1})=-1$, so $b_{1}=1/(f(y_{2})-y_{1})$ for some smooth real valued function $f$ of one variable and which cannot equal $y_{1}$ in a neighbourhood of $y_{*}$ (i.e. $f$ takes values in ${\mathbb{R}}$ minus a neighbourhood of $(y_{*})_{1}$). Actually the two cases can be joined in one case if we allow $f$ to be a smooth function taking values in $\hat{\mathbb{R}}$ and let $f(y_{2})=\infty$ iff $b_{1}(y_{1},y_{2})=0$ (recall that we work locally). Indeed if $b_{1}(y_{*})=0$ then we can locally write $b_{1}=h/(1-y_{1}h)$ for a smooth $h(y_{1},y_{2})$, which will play the role of $1/f$, but we have to prove that $h$ is independent of $y_{1}$. The equation $\partial_{1}b_{1}=b_{1}^{2}$ reads \big{(}(1-y_{1}h)\partial_{1}h+(h+y_{1}\partial_{1}h)h\big{)}/(1-y_{1}h)^{2}=h^{2}/(1-y_{1}h)^{2}. It simplifies into $\partial_{1}h=0$, which proves the claim. We can thus take $f=1/h$.

A similar analysis holds for $b_{2}$: $b_{2}=1/(g(y_{1})-y_{2})$ for some smooth $g$ defined locally, taking values in $\hat{\mathbb{R}}$, and avoiding the value $y_{2}$ in a neighbourhood of $y_{*}$.

Conversely taking such $f$ and $g$ defines smooth real valued coefficients $b_{1}$ and $b_{2}$ which satisfy the required differential equations. ∎

4.4.2. Signature $(2,0)$

In the case of signature $(2,0)$, the coefficients $a_{1}$ and $a_{2}$ can be given explicit expressions too, this time in terms of the natural complex structure of the chart.

Proposition 4.12**.**

We consider a connection $\nabla$ defined on a connected open subset $U$ of $E^{2,0}$ identified with ${\mathbb{C}}$. We assume that $\nabla$ is symmetric and projectively flattened and denote $a=a_{1}dx_{1}+a_{2}dx_{2}$ the associated form (see Section 3.1 and Definition 3.3). Then $\nabla$ is weakly stiff if and only if one of the following statements holds:

• •

$\nabla$* is trivial on $U$ or*

• •

the complex-valued function defined by $A=a_{1}+ia_{2}$ is of the form $A(z)=\frac{-2}{\bar{z}+f(z)}$ where $f$ is a meromorphic function such that $f(z)+\bar{z}$ does not vanish.

Conversely, any such meromorphic function $f$ defines on its domain, via $a_{1}(z)+ia_{2}(z)=-2/(\bar{z}+f(z))$, a weakly stiff connection for the canonical Euclidean structure of ${\mathbb{C}}$.

Proof.

Lemma 4.1 together with equation (3.4) shows that $\nabla$ is weakly stiff if and only if the coefficients $a_{1},a_{2}$ satisfy the system of equations:

1. (1)

$\partial_{1}a_{2}+\partial_{2}a_{1}=2a_{1}a_{2}$; 2. (2)

$\partial_{1}a_{1}-a^{2}_{1}=\partial_{2}a_{2}-a^{2}_{2}$.

We have $A^{2}=(a_{1}+ia_{2})^{2}=a^{2}_{1}-a^{2}_{2}+2ia_{1}a_{2}=(\partial_{1}+i\partial_{2})(a_{1}+ia_{2})$. The latter equation encompasses the two equations of the original system and reads

[TABLE]

Near points for which $A\neq 0$, this is equivalent to

[TABLE]

i.e. to $\bar{\partial}\left(\frac{1}{A}+\frac{\bar{z}}{2}\right)=0$. It follows that, locally, $A$ is either zero or a function of the form $\frac{-2}{\bar{z}+f(z)}$ where $f$ is a holomorphic function. Since $1/A$ does not vanish, $\bar{z}+f(z)$ does not either.

Near a point $z_{0}$ for which $A(z_{0})=0$, we consider the inverse of the quantity $-f$, i.e. we let $g=\frac{A}{2+\bar{z}A}$. For a $C^{1}$ complex valued function $A$, we have $\bar{\partial}g=\frac{2\bar{\partial}A-A^{2}}{\left(2+\bar{z}A\right)^{2}}$. The equation $A^{2}=2\bar{\partial}A$ is then equivalent to $\bar{\partial}g=0$, i.e. to $g$ being holomorphic. Hence either $A=0$ near $z_{0}$ or [math] is an isolated zero of $g$ and $f=-1/g$ is meromorphic.

Conversely if $a_{1}=a_{2}=0$, then the first-order infinitesimal holonomy matrices are equal to zero in the standard system of coordinates and $\nabla$ is flat. Besides, if $A=-2/(\bar{z}+f(z))$ for some meromorphic function such that $f(z)+\bar{z}\neq 0$ then $\bar{\partial}(1/A)=-1/2$, hence $A^{2}=2\bar{\partial}A$ at every point where $f\neq\infty$.

Near a point $z_{0}$ where $f=\infty$, which must be isolated, $A(z)=-\frac{2f^{-1}(z)}{1+\bar{z}f^{-1}(z)}$ where $f^{-1}$ vanishes at $z_{0}$. It follows that $A$ is smooth near $z_{0}$ and by continuity, $A^{2}=2\bar{\partial}A$ also holds at $z_{0}$. ∎

4.4.3. Unifying the two cases

Complex numbers are naturally attached to $E^{2,0}$ and $E^{0,2}$.

One can also attach a commutative ${\mathbb{R}}$-algebra ${\mathbb{A}}$ of dimension $2$ to $E^{1,1}$, defined by ${\mathbb{A}}={\mathbb{R}}\oplus\hat{\imath}{\mathbb{R}}$ and ${\hat{\imath}}^{2}=1$. Then $(u_{1}+u_{2}\hat{\imath})\times(v_{1}+v_{2}\hat{\imath})=(u_{1}v_{1}+u_{2}v_{2})+(u_{1}v_{2}+u_{2}v_{1})\hat{\imath}$. If we let $\bar{u}=u_{1}-u_{2}\hat{\imath}$ then $u\bar{u}=u_{1}^{2}-u_{2}^{2}$. We can then use any orthonormal777An orthonormal basis of a pseudo-Euclidean space is a basis whose vectors are of norm $\pm 1$ and are pairwise orthogonal. basis $(u,v)$ of $\vec{E}^{1,1}$ with $q(u)=1$ and $q(v)=-1$ to identify $\vec{E}^{1,1}$ with ${\mathbb{A}}$.

If we choose the canonical basis of $\vec{E}^{1,1}$ and perform the change of variable $x\mapsto y$ sending $E^{1,1}$ to $E^{\prime}$ then the product law of the algebra thus obtained on ${\mathbb{R}}^{2}$, call it ${\mathbb{A}}^{\prime}$, becomes extremely simple: $(a_{1},a_{2})\times(b_{1},b_{2})=(a_{1}b_{1},a_{2}b_{2})$, i.e. we have the Cartesian algebra product ${\mathbb{A}}^{\prime}={\mathbb{R}}\times{\mathbb{R}}$.

The analogue of a holomorphic function between (possibly identical) pseudo-Euclidean spaces of the same signature is a differentiable function from an open subset $U$ to the space, whose differential is everywhere a similarity for $q$ or [math]. In the ${\mathbb{A}}^{\prime}$ coordinates, this translates as a function $f:U\subset{\mathbb{A}}^{\prime}\to{\mathbb{A}}^{\prime}$ such that $\partial_{2}f_{1}=0$, $\partial_{1}f_{2}=0$. In other words, $f(y_{1},y_{2})=(f_{1}(y_{1}),f_{2}(y_{2}))$. Moreover, an element $y\in{\mathbb{A}}^{\prime}$ is invertible if and only if $y_{1}$ and $y_{2}$ are non-zero, and then $1/y=(1/y_{1},1/y_{2})$. Complete ${\mathbb{A}}^{\prime}\subset{\mathbb{R}}^{2}$ into $\hat{\mathbb{R}}\times\hat{\mathbb{R}}$. Then the function $y\mapsto 1/y$ is an involution of the completion, defined everywhere. An ${\mathbb{A}}^{\prime}$-meromorphic function is a function from an open subset of ${\mathbb{A}}^{\prime}$ to $\hat{\mathbb{R}}\times\hat{\mathbb{R}}$ of the form $f(y)=(f_{1}(y_{1}),f_{2}(y))$.

Also, in ${\mathbb{A}}^{\prime}$, $\bar{y}=(y_{2},y_{1})$, $1=(1,1)$ and the scalar product associated to $q(y)=y_{1}y_{2}=y\bar{y}$ is expressed as $(a\cdot b)=(a_{1}b_{2}+a_{2}b_{1})/2$. In particular, in ${\mathbb{A}}^{\prime}$, the $1$-form $b\mapsto(a\cdot b)$ associated to $a=a_{1}e_{1}+a_{2}e_{2}$ is equal to $\frac{a_{2}}{2}e_{1}^{*}+\frac{a_{1}}{2}e_{2}^{*}$. Conversely to a $1$-form $a=a_{1}e_{1}^{*}+a_{2}e_{2}^{*}$ one associates the vector $2a_{2}e_{1}+2a_{1}e_{2}$.

More generally to any pseudo-Euclidean space $E$ of dimension two and signature $(2,0)$, $(1,1)$ or $(0,2)$, together with a choice of orthonormal basis, one associates a product turning $\vec{E}$ into an ${\mathbb{R}}$-algebra, and a notion of $\vec{E}$-meromorphic function, which is independent of the choice of orthonormal basis.

We can then unify and extend Propositions 4.11 and 4.12 into:

Proposition 4.13**.**

Consider a pseudo-Euclidean space $E$ of dimension $2$ and any choice of associated algebra structure on $\vec{E}$ as described above. Consider a connection $\nabla$ defined on an open subset of $E$, such that $\nabla$ is symmetric and projectively flattened. Denote $a$ the associated form (see Section 3.1 and Definition 3.3). Let $A$ be the vector field obtained from $a$ by the musical isomorphism888$a\{x\}(u)=(A(x)\cdot u)$ and consider $A$ as a function from $E$ to $\vec{E}$. Then $\nabla$ is weakly stiff for $E$ if and only if

[TABLE]

for the notion of inverse in the algebra, where $f$ is $E$-meromorphic (and smooth) and $1/(\bar{x}+f(x))$ remains finite for all $x$.

Conversely, any $A$ thus obtained defines, via the associated form $a$, a weakly stiff connection for $E$.

4.5. Canonical models of stiff connections

We recall that we denote by $q(x)$ the standard pseudo-Euclidean squared norm of any vector $x\in E^{p,d-p}$, i.e.

[TABLE]

Recall that the potential of a projectively flattened connection is only unique up to multiplication by a non-zero real constant.

For each possible signature, we define here canonical models for pseudo-Euclidean charts, corresponding to potentials $\psi(x)=q(x)+\lambda$ where $\lambda\in{\mathbb{R}}$ (this corresponds to

[TABLE]

in Theorem 4.4). We will have two settings. One in which we will allow changes of variables that are similarities. Then we impose $\lambda\in\{{-1,0,1\}}$. The other one, less frequent, where we only allow isometries. Then we allow any value of $\lambda\in{\mathbb{R}}$.

The connection is only defined on the set of $x\in E^{p,d-p}$ where $\psi(x)\neq 0$. For a given potential $\psi$, we may have more than one connected components of the domain where $\psi\neq 0$. We recall without proof the following basic facts in the next two lemmas, that the reader can skip on first reading:

Lemma 4.14**.**

We assume $d\geq 1$. Let $E=E^{p,d-p}$. Let

[TABLE]

• •

$U^{+}$* is empty if and only if $p=d$ and $\lambda\leq 0$.*

• •

$U^{-}$* is empty if and only if $p=0$ and $\lambda\geq 0$.*

• •

$U^{+}$* contains [math] if and only if $\lambda>0$.*

• •

$U^{-}$* contains [math] if and only if $\lambda<0$.*

• •

If $U^{\pm}$ contains [math] then it deformation retracts to [math]. Otherwise it deformation retracts to a subset homeomorphic to the unit sphere $S^{n-1}$ of the Euclidean space of dimension $n=p$ for $U^{+}$ or $n=p-d$ for $U^{-}$.

As a consequence

• •

$U^{+}$* is disconnected if and only if $p=1$ and $\lambda\leq 0$. In this case it consists in two components $U^{+}_{L}$ and $U^{+}_{R}$ defined by $x_{1}>0$ and $x_{1}<0$.*

• •

$U^{-}$* is disconnected if and only if $d-p=1$ and $\lambda\geq 0$. In this case it consists in two components $U^{-}_{L}$ and $U^{-}_{R}$ defined by $x_{d}>0$ and $x_{d}<0$.*

• •

Both $U^{+}$ and $U^{-}$ are disconnected if and only if the signature is $(1,1)$ and $\lambda=0$,

• •

When connected, $U^{+}$ is simply connected unless $p=2$ and $\lambda\leq 0$. When disconnected, its two components are simply connected.

• •

When connected, $U^{-}$ is simply connected unless $q=2$ and $\lambda\geq 0$. When disconnected, its two components are simply connected.

Moreover (with the convention that the empty set is bounded)

• •

$U^{+}$* is bounded if and only if $p=0$.*

• •

$U^{-}$* is bounded if and only if $p=d$.*

• •

$U^{+}=E$* if and only if $p=d$ and $\lambda>0$.*

• •

$U^{-}=E$* if and only if $p=0$ and $\lambda<0$.*

We use the standard notation $\operatorname{O}(p,d-p)=\operatorname{O}(E)$ for the group of isometries of $E$ fixing [math] (a.k.a. generalized orthogonal group), $\operatorname{SO}(p,d-p)=\operatorname{SO}(E)$ the subgroup of those which are direct (orientation preserving). Maybe less known is the subgroup $\operatorname{O}^{+}(E)$ of elements of $\operatorname{O}(E)$ preserving the orientation of every subspace of dimension $p$ on which $q$ is positive definite. These isometries are called orthochronous. They are those whose matrix in the canonical basis of $E^{p,d-p}$ have their upper left $(p,p)$ sub-matrix of positive determinant. By analogy, we denote $\operatorname{O}^{-}(E)$ the isometries preserving the orientation of every subspace of dimension $d-p$ on which $q$ is negative definite. Of course in signature $(d,0)$ we have $\operatorname{O}^{-}(E)=\operatorname{O}(E)$ and $\operatorname{O}^{+}(E)=\operatorname{SO}(E)$. And in signature $(0,d)$ we have $\operatorname{O}^{-}(E)=\operatorname{SO}(E)$ and $\operatorname{O}^{+}(E)=\operatorname{O}(E)$.

Lemma 4.15**.**

We assume $d\geq 2$. For a subset $U$ of $E$ and a subgroup $H$ of $O(E)$ we denote $H(U)$ the subgroup of $\operatorname{O}(E)$ of elements preserving the set $U$.

• •

The group $\operatorname{SO}_{0}(E)=\operatorname{O}^{-}(E)\cap\operatorname{O}^{+}(E)$ is always connected.

• •

$\operatorname{O}(U^{+})=\operatorname{O}(U^{-})=\operatorname{O}(E)$.

• •

If $U^{+}$ is disconnected then: $\operatorname{O}(U^{+}_{L})=\operatorname{O}(U^{+}_{R})=\operatorname{O}^{+}(E)$.

• •

If $U^{-}$ is disconnected then: $\operatorname{O}(U^{-}_{L})=\operatorname{O}(U^{-}_{R})=\operatorname{O}^{-}(E)$.

• •

On each connected component $U$ of $U^{+}$ or $U^{-}$, the group $\operatorname{SO}_{0}(U)$ acts transitively on each level set \cal L_{a}(U)=\big{\{}x\in U\,;\,q(x)=a\big{\}}.999They are allowed to be empty: any group action on the empty set is transitive. These level sets are connected.

• •

*The group $\operatorname{O}(E)$ acts transitively on \cal L^{*}_{0}=\big{\{}x\in E\,;x\neq 0\text{ and }\,q(x)=0\big{\}}.101010These sets may or may not be connected. We do not indicate here when this is the case. *

Definition 4.16** (Canonical models).**

In the pseudo-Euclidean space $E^{p,d-p}$ endowed with the canonical form $q(x)=x_{1}^{2}+\cdots+x_{p}^{2}-x_{p+1}^{2}-\cdots-x_{d}^{2}$, let $\nabla$ be the projectively flattened connection of potential $\psi(x)=q(x)+\lambda$. We denote

[TABLE]

the restriction of $(E,\nabla)$ to the set of $x\in E$ such that $\psi(x)>0$ (i.e. the set denoted $U^{+}$ in the above lemma), and restricted furthermore to $x_{1}>0$ if $p=1$ and $\lambda\leq 0$ (i.e.  $U^{+}_{R}$).

Similarly, we denote

[TABLE]

the model restricted to $\psi<0$, and restricted furthermore to $x_{d}>0$ if $q=1$ and $\lambda\geq 0$.

In particular the canonical models are defined on connected components of the set \big{\{}x\in E\,\;\,\psi\neq 0\big{\}}.

For signatures of the form $(d,0)$ and constants $\lambda\geq 0$, the set underlying $\mathcal{S}_{\lambda}^{-}(d,0)$ is empty, and we exclude this situation from the collection of models. Similarly, for signatures of the form $(0,d)$ and constants $\lambda\leq 0$, the set underlying $\mathcal{S}_{\lambda}^{+}(0,d)$ is empty, and we also exclude this from the collection of models. In all the other cases the model is non-empty and connected. The models of signature $(2,0)$ are illustrated in Figure 1 and for signature $(1,1)$ in Figure 2.

The function $\psi$ is invariant by the group $\operatorname{O}(E)$, and Corollary 3.14 implies that $\nabla$ is invariant by this group. For $\nu\in\{+,-\}$ and any $\lambda\in{\mathbb{R}}$, the group preserves the domain of $\cal S^{\nu}_{\lambda}(p,d-p)$ except in the cases where we had to restrict to $x_{1}>0$ or $x_{d}>0$: here we have to take the index $2$ subgroup $\operatorname{O}(U^{\nu}_{R})=\operatorname{O}^{\nu}(E)$ described in Lemma 4.15. Also, models $\mathcal{S}_{0}^{\pm}(p,d-p)$ are invariant by rescalings $x\mapsto rx$ with $r>0$.

Remark 4.17*.*

There are correspondence between some models. For instance, there is a linear bijection between $\mathcal{S}_{\lambda}^{+}(p,d-p)$ and $\mathcal{S}_{-\lambda}^{-}(d-p,p)$ that multiplies $q$ by $-1$. It is not a similarity.

Remark 4.18*.*

We introduced the scalar curvature of $\nabla$ relative to $g$ near the end of Section 4.2 and denoted it $S(\nabla,g)$. We determined in Remark 4.9 that its sign is the sign of $\psi$. We immediately get that, on $\cal S^{\nu}_{\lambda}(p,d-p)$, it is $\nu$.

Let us show that every non-flat stiff connection is locally equivalent to restrictions of the canonical models.

Proposition 4.19**.**

We consider a pair $(M,\nabla)$ of a similarity pseudo-Euclidean manifold $M$ and a stiff connection $\nabla$ on it. Then near every point $P$ of $M$ either $\nabla$ is locally flat or (and these two cases are mutually exclusive) there is a local chart of $(M,\nabla)$ equivalent to a restriction of a unique canonical model $\mathcal{S}_{\lambda}^{\nu}(p,d-p)$ with $\nu\in\{-,+\}$ and $\lambda\in\{-1,0,1\}$.

Proof.

Consider any local pseudo-Euclidean chart $\phi$ and let $x_{*}=\phi(P)\in E$. We identify $E$ with the vector space ${\mathbb{R}}^{d}$. Following Theorem 4.4, locally $\psi(x_{1},\dots,x_{d})=\frac{K}{2}q(x)+\Lambda(x_{1},\dots,x_{d})$ for some $K\in{\mathbb{R}}$ and $\Lambda$ an affine map $E\to{\mathbb{R}}$.

By Remark 4.6, if $K=0$ then the connection is flat near $P$ (and conversely).

Otherwise, we will make use of Corollary 3.14 and of the fact that any non-zero scalar multiple of $\psi$ yields the same connection. Denote $x\cdot y$ the scalar product of vectors $x$ and $y$ associated to the pseudo-Euclidean norm, so $q(x)=x\cdot x$. Since it is nondegenerate, there exists $v\in E$ and $k\in{\mathbb{R}}$ such that for all $x$, $\Lambda(x)=x\cdot v+k$. Then $\psi(x-\frac{1}{K}v)=\frac{K}{2}q(x)+k$. Let $r>0$. Then $\psi(rx-\frac{1}{K}v)=\frac{Kr^{2}}{2}x\cdot x+k$ hence

[TABLE]

with $\lambda=\frac{2k}{Kr^{2}}$. So according to the sign of $k/K$, we can choose $r$ so that $\lambda\in\{-1,0,1\}$ and the change of variable $x\mapsto rx-v$ gives the new chart in which $\nabla$ is a restriction of a canonical model. Since $q(x)+\lambda$ cannot vanish on the chart, its sign $\nu$ is locally constant and determines, together with $\lambda$, the canonical model among the six (or four) possible choices for the given signature. In the case where the domain on which111111Where $\nu=-$ is replaced by $-1$ and $\nu=+$ by $1$. $\nu\psi>0$ is not connected, a further change of variable may be needed if we are in the wrong component. If we do not care about orientability one can take $x\mapsto-x$. If we care, since $d\geq 2$, we can take for instance the map that negates $x_{1}$ and $x_{d}$ and leaves other coordinates invariant.

No affine bijection can change the degree of $\psi$, hence the first and second case are locally incompatible.

Uniqueness comes from Lemma 4.22 in Section 4.7. ∎

Remark 4.20*.*

If we only allow for changes of variables that are isometries of $E$ instead of similarities, then we can only reduce to $\psi=q(x)+\lambda$ where $\lambda=2k/K$ can be any real. We thus get a one-parameter family, parameterized by the “squared radius” $-\lambda$ (hyphens come in because this quantity can as well be negative, [math], or positive) of the pseudo-Euclidean sphere of equation $\psi(x)=0$.

4.6. Slicing canonical models and the sign of curvature

If we consider a canonical model and cut it by an affine subspace $F$ of dimension at least two, intersecting the domain $U$ of the model, and such that the restriction of $q$ to $F$ is nondegenerate, then one obtains a model isomorphic to one of the canonical models.

Indeed by Proposition 3.16 the connection has a restriction $\nabla^{F}$ to $F\cap U$ and the restriction of $\psi=q+\lambda$ to $F$ is a potential for $\nabla^{F}$. By nondegeneracy, $U\cap F$ and $\psi|_{F}$ can be mapped to a canonical model by some similarity/isometry.

Interestingly, all canonical models for all possible signatures in dimension $d$ can be obtained as slices of a single pseudo-Euclidean space $E^{d+1,d+1}$ with the connection $\nabla$ associated to $\psi=q+0$ and the two models it contains: $\cal S^{+}_{0}(d+1,d+1)$ and $\cal S^{-}_{0}(d+1,d+1)$.

We saw at the end of Section 4.2 that the sign of the scalar curvature of $\nabla$ relative to $g$, $S(\nabla,g)$, is the same on the submodel. This sign is given by $\nu$ since $S(\nabla,g)=d(d-1)2/\psi$. On any 2 dimensional affine plane $P$ through a point $x$ and such that the restriction of $q$ to $\vec{P}$ is nondegenerate, the interpretation of the sign of scalar curvature given near the end of Section 4.2 can be rephrased as follows: an infinitesimal loop based on $x$ is an infinitesimal isometry pushing $v$ in the same sense as the loop if $\nu q(v)>0$ and in the opposite sense if $\nu q(v)<0$.

4.7. Local isomorphisms between canonical models

We introduce the following terminology for practicality in the next statement, without intention to see it become a standard.

Definition 4.21**.**

Let $E$ and $E^{\prime}$ are two pseudo-Euclidean spaces of the same dimension but not necessarily the same signature. Denote $q$ and $q^{\prime}$ the respective quadratic forms on $E$ and $E^{\prime}$. A negalitude is an affine bijection $f$ such that $\forall x\in\vec{E}$, $q^{\prime}(\vec{f}(x))=-r^{2}q(x)$, where the constant $r>0$ is called its ratio. We designate the set of these maps by $\cal N$.

Lemma 4.22**.**

Consider two models $\cal S=\cal S^{\nu}_{\lambda}(p,d-p)$ and $\cal S^{\prime}=\cal S^{\nu^{\prime}}_{\lambda^{\prime}}(p^{\prime},d-p^{\prime})$ (note that the signatures are not assumed identical) with $\lambda,\ \lambda^{\prime}\in{\mathbb{R}}$. Denote $\nabla$, $\nabla^{\prime}$ the respective connections. Assume that $U$ and $U^{\prime}$ are connected open subsets of models $\cal S$ and $\cal S^{\prime}$, that $f:U\to U^{\prime}$ is a diffeomorphism and that, on $U$, $f_{*}\nabla=\nabla^{\prime}$. Then either:

1. (1)

$f$* is the restriction of a similarity and*

• •

$p^{\prime}=p$,

• •

$f$* fixes [math],*

• •

$\lambda^{\prime}=\lambda r^{2}$* where $r>0$ is the scaling factor of $f$,*

• •

$\nu^{\prime}=\nu$.

In particular $\lambda$ and $\lambda^{\prime}$ have the same sign and if they are both of absolute value $1$, then they are equal and $f$ is an isometry. 2. (2)

Or $f$ is the restriction of a negalitude and

• •

$p^{\prime}=d-p$,

• •

$f$* fixes [math],*

• •

$\lambda^{\prime}=-\lambda r^{2}$* where $r>0$ is the scaling factor of $f$,*

• •

$\nu^{\prime}=-\nu$.

Proof.

We first prove that $f$ is the restriction of an affine map. (In many applications of the lemma, we will already know that this is the case.) Note that $f_{*}\nabla=\nabla^{\prime}$ implies that $f$ sends geodesics to geodesics, and since the geodesics of the canonical models are straight lines and $d\geq 2$, a classical result (see [Löw82]) tells us that $f$ is the restriction of a projective mapping. Write $f=F/C$ where $F:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ and $C:{\mathbb{R}}^{d}\to{\mathbb{R}}$ are affine. Proposition 3.15 then shows that $\psi/C$ and $\psi^{\prime}$ are locally proportional and by connectedness the proportionality constant is the same on all $U$ (recall that potentials do not vanish). So $\exists\alpha\in{\mathbb{R}}^{*}$ such that

[TABLE]

This is only possible if $C$ is constant: indeed the equation reads $q^{\prime}(F/C)+\lambda^{\prime}=\alpha\frac{q+\lambda}{C}$, and since $q^{\prime}(F/C)=q^{\prime}(F)/C^{2}$ it is equivalent to $q^{\prime}(F)+\lambda^{\prime}C^{2}=(q+\lambda)\alpha C$ and if $C$ were not constant, the left hand side would have degree $2$ whereas and the right hand side would have degree $3$. Thus $C$ is constant and $f$ is the restriction of an affine bijection,

By reducing $U$ we may assume that it is connected. By Corollary 3.14 and uniqueness of $\psi$ up to multiplication by a non-zero constant, the following holds on $U$: $\psi^{\prime}\circ f=\alpha\psi$ for some $\alpha\in{\mathbb{R}}^{*}$, i.e. $q\circ f+\lambda^{\prime}=\alpha\times(q^{\prime}+\lambda)$. Let us write $f(x)=\ell(x)+c$ where $\ell$ fixes [math]. Hence, denoting $u\cdot v$ the scalar product of $E^{p,d-p}$, and $u\cdot^{\prime}v$ the scalar product of $E^{p^{\prime},d-p^{\prime}}$,

[TABLE]

which holds on an open set and develops to

[TABLE]

Identifying the terms with the same degree we get

[TABLE]

Since the scalar product is nondegenerate we have $c=0$, so $\lambda^{\prime}=\alpha\lambda.$ The maps $q^{\prime}\circ\ell$ and $\alpha q$ have the same signature.

If $\alpha>0$ this implies $p^{\prime}=p$. Moreover $\ell/\sqrt{\alpha}$ is an isometry and $\ell$ a similarity of ratio $\sqrt{\alpha}$. Finally $\lambda^{\prime}=\alpha\lambda=r^{2}\lambda$.

If $\alpha<0$ this implies $p^{\prime}=d-p$ and $\ell$ is is a negalitude of ratio $r>0$ such that $\alpha=-r^{2}$. Moreover $\lambda^{\prime}=\alpha\lambda=-r^{2}\lambda$.

The symbol $\nu$ indicates the sign of $\psi(x)$ for $x\in U$ and $\nu^{\prime}$ the sign of $\psi^{\prime}\circ f(x)$ for $x\in U$. Since $\psi^{\prime}\circ f=\alpha\psi$, it follows that $\nu=\nu^{\prime}$ when $\alpha>0$ and $\nu=-\nu^{\prime}$ when $\alpha<0$. ∎

Remark 4.23*.*

Note that if $p=d-p$, then Case (2) of the lemma also satisfies $p^{\prime}=p$, i.e. $f$ goes from $E=E^{p,p}$ to itself. Actually, there does exist negalitudes from $E$ to itself. At the level of vectors, they exchange the positive and negative cones. Moreover, some or all negalitudes fixing [math] (depending on $\lambda$, $\nu$ and $p$) send the domain and connection of $\cal S^{\nu}_{\lambda}(p,p)$ to the domain and connection of $\cal S^{-\nu}_{-\lambda}(p,p)$; we do not call them isomorphisms because they do not respect the underlying (similarity) pseudo-Euclidean structure.

Remark 4.24*.*

Consider the connection $\nabla$ of one of the canonical models, but expressed, near a point, in a differentiable chart that is not a pseudo-Euclidean chart. It is possible to deduce, from $\nabla$ alone, which canonical model it came from, up to sign reversal. More precisely, uniqueness of the model up to sign reversal follows from Lemma 4.22. For a practical method to tell which model is the right one, one has to first find a chart that projectively flattens $\nabla$. Then one can apply the method of the proof of Lemma 4.22.

Definition 4.25**.**

By automorphism of a canonical model $\cal S$, we mean a bijection from $\cal S$ to itself, that is an isomorphism of pseudo-Euclidean similarity manifold, and preserves $\nabla$.

Lemma 4.26**.**

Consider a canonical model $\cal S$ and a map $\phi:U\to V$ between connected open subsets of $\cal S$. Assume that $\phi$ is an isomorphism for the pseudo-Riemannian similarity manifold structure that $\cal S$ induces on $U$ and $V$ and that it preserves $\nabla$. Then $\phi$ is the restriction of a similarity of $E$, and this similarity restricts to an automorphism of $\cal S$.

Proof.

A map such as $\phi$ must be locally the restriction of affine similarities of $E$. Since $U$ is connected, $\phi$ is the restriction of a single affine similarity $F$ of $E$. On $U$, the map $F$ preserves $\psi$ up to multiplication by a constant: $\psi\circ F=\alpha\psi$ for some $\alpha\in{\mathbb{R}}^{*}$. Since $\psi$ is polynomial and $F$ affine, the relation $\psi\circ F=\alpha\psi$ actually holds on all $E$. Since $\cal S$ is connected and $F$ maps at least one point of $\cal S$ in $\cal S$, we have $F(\cal S)\subset\cal S$. Since $F^{-1}$ maps at least one point of $\cal S$ in $\cal S$ we have $F^{-1}(\cal S)\subset\cal S$. ∎

Let $E=E^{p,d-p}$ and recall that we denote $\operatorname{O}(E)$ the group of isometries of $E^{p,d-p}$ fixing [math]. Just before Lemma 4.15 we defined the subgroups $\operatorname{O}^{-}(E)$ and $\operatorname{O}^{+}(E)$.

Proposition 4.27**.**

As pseudo-Euclidean similarity manifolds (case 1) or pseudo-Euclidean manifolds (case 2) endowed with a stiff connection, the automorphisms $\phi$ of the canonical models consist in:

• •

If we are in case 1 and $\lambda=0$: the similarities ${\mathbb{R}}^{*}\!.\!\operatorname{O}^{+}(E)$ if ($p=1$* and $\nu=+$), ${\mathbb{R}}^{*}\!.\!\operatorname{O}^{-}(E)$ if ($d-p=1$ and $\nu=-$), the whole group of similarities ${\mathbb{R}}^{*}\!.\!\operatorname{O}(E)$ otherwise.*

• •

Otherwise: the group $\operatorname{O}^{+}(E)$ if ($p=1$, $\nu=+$ and $\lambda<0$), $\operatorname{O}^{-}(E)$ if ($d-p=1$, $\nu=-$ and $\lambda>0$), the group $\operatorname{O}(E)$ otherwise.

Proof.

By Lemma 4.26, $\phi$ is the the restriction of affine similarities of $E$. Lemma 4.22 then proves that $\phi$ is of the wanted form.

Conversely, maps $\phi$ of this form are automorphisms of the canonical model, since they map the domain to itself and satisfy $\psi\circ\phi=r^{2}\psi$ for some $r>0$ (case 1) or $\psi\circ\phi=\psi$ (case 2). ∎

4.8. Zooming in and out

Recall that the models $\mathcal{S}_{0}^{\nu}(p,d-p)$, $\nu\in\{-,+\}$ are invariant by a linear rescaling $x\mapsto rx$. Consider any model $\mathcal{S}_{\lambda}^{\pm}(p,d-p)$ whose domain is unbounded and let us rescale, i.e. perform the change of variable $y=rx$ with $r\in{\mathbb{R}}*$ (we impose $r>0$ if the set where $\nu\psi>0$ is disconnected, see Definition 4.16). Initially, $\psi(x)=q(x)+\lambda$. We have seen that in the new variable, the connection $\nabla$ has potential $\psi(y/r)$, which is given by $q(y)/r^{2}+\lambda$. The connection is also given by $r^{2}$ times this potential, i.e. by $\psi_{r}:=q+\lambda r^{2}$, hence the model, $\cal S^{\nu}_{\lambda}(p,d-p)$, is mapped to model $\cal S^{\nu}_{\lambda r^{2}}(p,d-p)$.

Zooming out. When $r\longrightarrow 0$, $\psi_{r}$ tends uniformly on compact subsets of ${\mathbb{R}}^{d}$ to $\psi_{0}=q(x)$. Moreover the domain where121212See Footnote 11. $\nu\psi_{r}>0$ tends to the domain where $\nu\psi_{0}>0$ (and similarly if we impose $x_{1}>0$ or $x_{d}>0$ in the case those sets are disconnected, see Lemma 4.14). Hence in some sense, the $r$-rescaling of models $\mathcal{S}_{\lambda}^{\pm}(p,d-p)$ tend to $\mathcal{S}_{0}^{\pm}(p,d-p)$ or a finite union of copies thereof. In particular all unbounded models $\cal S^{\pm}_{\lambda}(p,d-p)$ are asymptotically rescaling-invariant at $\infty$.

Models $\mathcal{S}_{\lambda}^{-}(d,0)$ and $\mathcal{S}_{\lambda}^{+}(0,d)$ are either empty or defined on the open unit ball. It is thus pointless to consider their behaviour at infinity.

Zooming in. What about the limit $r\longrightarrow\pm\infty$? For $\lambda=0$, since the connection is rescaling-invariant, it is equal to its rescaling limit as $r\to\pm\infty$. For $\lambda\neq 0$, we of course obtain the trivial connection on $E$ if [math] is in the interior of the domain of the model (iff. $\nu\lambda>0$), as would hold for any smooth connection.

Deformations. We can give another point of view on the zooming-in procedure: for $\lambda\neq 0$, letting $r\longrightarrow\pm\infty$ amounts to considering model $\cal S^{\pm}_{\lambda}(p,d-p)$ and let $\lambda$ tend to $\pm\infty$. Recall that the potential $\psi$ is defined only up to a scaling. Hence a potential is also $1+\mu q$ with $\mu=1/\lambda$ and as $\mu\longrightarrow 0$, this potential tends to the constant $1$. On the domain $\nu q>0$, the associated form $a$ locally tends to be uniformly zero and $\nabla$ becomes the trivial connection. Hence can consider the family of connections $\nabla_{\lambda}$ of potential $\psi=q+\lambda$ as a family of deformations of the trivial connection.

Interestingly, we can also obtain all the projective transformations in ${\mathbb{R}}^{d}$ of the trivial connection as limits of properly rescaled and translated canonical models. Indeed, by Proposition 3.15 applied to $\tilde{\psi}=1$, the potential of such a connection can be written in the form $\psi=C$ where $C$ is the restriction of a non-zero affine map from ${\mathbb{R}}^{d}$ to ${\mathbb{R}}$. Define a potential $\psi_{\tau}(x)=\tau q(x)+C(x)$. Near every $x$ for which $C(x)\neq 0$, the function $\psi$ is non-zero when $\tau\in{\mathbb{R}}$ is small enough. By Proposition 4.19, the component containing $x$ of the subset of ${\mathbb{R}}^{d}$ equation $\psi_{\tau}\neq 0$ can be mapped isomorphically, as a pseudo-Euclidean similarity surfaces with a connection, to one of the canonical models.

5. Intermezzo: conformal considerations

In this section we consider variations of the constrains on $\nabla$ and their consequences. This in particularly shows the relevance of the infinitesimally conformal hypothesis.

5.1. Infinitesimal homotheties

We may wonder if there are non-flat symmetric connections that are projectively flat and whose infinitesimal holonomies are all homotheties. The following lemma proves that this is not the case.

Proposition 5.1**.**

Let $\nabla$ be a symmetric and projectively flattened connection on an open subset $U$ of $\mathbb{R}^{d}$ whose infinitesimal holonomies are all infinitesimal homotheties. Then $\nabla$ is flat. It is trivialized on each connected component of $U$ by a projective transformation.

Proof.

Volume-preserving homotheties are trivial so the infinitesimal holonomy of $\nabla$ is trivial if $\nabla$ is incompressible. If $d\geq 3$, then Proposition 4.10 proves that $\nabla$ is automatically incompressible. It remains thus to consider the case where $d=2$.

Applying Equation (3.4) we obtain that the coefficients $R_{12k}^{l}=-M(1,2)_{k,l}$ of the opposite of the first order infinitesimal holonomy matrix $M(1,2)$ on an infinitesimal rectangle directed by $e_{1}$ and $e_{2}$ are:

• •

$R_{121}^{1}=-2\partial_{2}a_{1}+\partial_{1}a_{2}+a_{1}a_{2}$;

• •

$R_{121}^{2}=\partial_{1}a_{1}-a_{1}^{2}$;

• •

$R_{122}^{1}=-\partial_{2}a_{2}+a_{2}^{2}$;

• •

$R_{122}^{2}=2\partial_{1}a_{2}-\partial_{2}a_{1}-a_{1}a_{2}$.

Since these matrices are of the form $\lambda I$ for some $\lambda\in\mathbb{R}$, the coefficients $a_{1}$ and $a_{2}$ satisfy the following equations:

1. (1)

$\partial_{1}a_{2}+\partial_{2}a_{1}=2a_{1}a_{2}$; 2. (2)

$\partial_{1}a_{1}=a_{1}^{2}$; 3. (3)

$\partial_{2}a_{2}=a_{2}^{2}$.

Let us introduce the quantity $D=\partial_{1}a_{2}-a_{1}a_{2}$. A direct computation proves that $\partial_{1}D=3a_{1}D$ and $\partial_{2}D=3a_{2}D$. We first assume $D$ does not vanish at some point, we obtain that locally $a_{1}$ and $a_{2}$ respectively coincide with $\partial_{1}E$ and $\partial_{2}E$ where $E=\frac{\ln|D|}{3}$. This implies $\partial_{2}a_{1}=\partial_{1}a_{2}$. The first equation then implies $D=0$ at this point. Consequently, we have $D=0$ and $a_{1}a_{2}=\partial_{1}a_{2}$.

The first equation then also implies that $\partial_{2}a_{1}=a_{1}a_{2}$. The holonomy matrix is thus equal to zero and the curvature tensor $R$ vanishes. It follows that $\nabla$ is flat in the domain of $\phi$. Flat symmetric connections are trivializable. Any trivialization must locally preserve straight lines and hence be locally projective ([Löw82]). ∎

5.2. Conformality for projectively flattened connections

Given a pseudo-Euclidean space $E$ of signature $(p,d-p)$ and a connection $\nabla$ defined on an open subset $U$ of $E$, we say that $\nabla$ is weakly conformal in the chart whenever we can cover $U$ with open sets on which all loops define holonomies that preserve the conformal structure of $E$, i.e. they are similarities of $E$. This is a local condition. We say that $\nabla$ is conformal in the chart whenever the parallel transport preserves the conformal structure of $E$, i.e. it consists only of similarities of $E$. We have:

conformal $\implies$ weakly conformal $\implies$ infinitesimally conformal.

Remark 5.2*.*

A connection may be weakly conformal but fail to be conformal: if one transports the canonical connection by a diffeomorphism between subsets of ${\mathbb{R}}^{d}$, one obtains most of the time a connection for which the parallel transport does not consist only in similarities of ${\mathbb{R}}^{d}$, though all holonomies are the identity, hence similarities. Moreover, if the diffeomorphism is projective with non-constant denominator, we get an example where the connection is projectively flattened.

Proposition 5.3**.**

Assume that $\nabla$ is projectively flattened, and weakly conformal for $E$. Then $\nabla$ is flat.

Proof.

Without loss of generality we assume that $\nabla$ is defined on a connected domain.

We start by the case where $\nabla$ is conformal, i.e. its parallel transport is conformal. In particular, perpendicular vectors remain perpendicular. Then if we consider the holonomy of the connection along a rectangle formed two segments directed by vectors $e_{i},e_{j}$ of the canonical basis, the parallel transport of a vector in direction $e_{i}$ or $e_{j}$ remains in the plane spanned by $e_{i}$ and $e_{j}$ (see Corollary 3.6). Besides, by the projective hypothesis, while moving in direction $e_{i}$, the vector $e_{i}$ remains parallel to itself, and since conformal maps preserve orthogonality, $e_{j}$ remains parallel to itself too. The same is true if the movement is in the direction $e_{j}$. Thus, the associated holonomy preserves the direction of every base element $e_{i}$, and since it is conformal, it is a homothety. It follows that first-order infinitesimal holonomies are all homotheties. Proposition 5.1 then implies that the connection is flat.

In the general case, where we only assume $\nabla$ weakly conformal, we distinguish several cases. First note that $\nabla$ is infinitesimally conformal.

If $d\geq 3$ then we saw in Proposition 4.10 that the connection is actually stiff, and Theorem 4.4 thus applies. If $K=0$ then the connection is flat, by Remark 4.6. If $K\neq 0$ then we will see in Section 6.4 that on any neighbourhood of a point $x$, the holonomy group at $x$ is $\operatorname{SL}(d,{\mathbb{R}})$ (identifying $\vec{E}$ to ${\mathbb{R}}^{d}$), contradicting the hypothesis.

If $d=2$ then consider a point in $x$. For an open subset $V$ of $U$ and a point $p\in v$, denote $\operatorname{Hol}(p,V)$ the group of all holonomies of loops contained in $V$ and based at $p$. By hypothesis, $x$ has an open neighbourhood $V$ for which $\forall p\in V$, $\operatorname{Hol}(p,V)$ is contained in the group of similarities of $\vec{E}$. Without loss of generality we may assume $V$ connected. Note that parallel transport along a path in $V$ defines a conjugacy between the two groups $\operatorname{Hol}(p,V)$ at the two ends $p$ of the path.

We first assume that $\operatorname{Hol}(x,V)$ only consists in homotheties. Then this is the case for all $\operatorname{Hol}(p,V)$ where $p\in V$, because the conjugate of a homothety is a homothety. Then the infinitesimal holonomy of every point in $V$ also only consists in homotheties and by Proposition 5.1, $\nabla$ is flat.

Second, we treat the case where $\operatorname{Hol}(x)$ does not only consist only in homotheties. Then for the same reason, this is not the case either for any $\operatorname{Hol}(p,V)$ where $p\in V$. Consider then the parallel transport along any path in $V$. It is an orientation preserving linear map $L$ between respective tangent spaces, conjugating local holonomies to local holonomies. In particular it conjugates at least one conformal map of $\vec{E}$ that is not a homothety, to a conformal map of $\vec{E}$. Such a map $L$ is necessarily131313It is a simple exercise in algebra, using that $(2,0)$-conformal maps have matrices of the form $\begin{bmatrix}a&b\\ -b&a\end{bmatrix}$. conformal in the $(2,0)$ and $(0,2)$ signatures. For signature $(1,1)$, in model $E^{\prime}$ where $q=y_{1}y_{2}$, one can prove141414In $E^{\prime}$, conformal linear maps take the form $\begin{bmatrix}a&0\\ 0&b\end{bmatrix}$ with $ab>0$. that the map $L$ is either $E^{\prime}$-conformal or the composition of $(y_{1},y_{2})\mapsto(-y_{2},y_{1})$ and an $E^{\prime}$-conformal map. By slicing a path in small parts, each corresponding $L$ is close to the identity hence cannot be of the latter type. Then $L$ is a composition of conformal maps so it is conformal. We conclude using the first part of the proof. ∎

Remark 5.4*.*

For the standard sphere in ${\mathbb{R}}^{3}$, the chart obtained by central projection to a plane $P$ not passing through the centre is a projective flattening: it sends great circles to straight lines. However, the Levi-Civita connection of the standard spherical metric, which is in particular a conformal connection for the spherical metric, is not even infinitesimally conformal for the conformal structure of the plane $P$.

6. Parallel transport in canonical models

In any canonical model, the connection $\nabla$ is defined by a potential of the form $\psi(x)=q(x)+\lambda$ where $q(x)$ is quadratic form associated to the pseudo-Euclidean structure. Let $x\cdot y$ denote the associated scalar product of two vectors $x$ and $y$. In particular, $q(x)=x\cdot x$. If $E=E^{p,d-p}$ then $q(x)=x_{1}^{2}+\cdots+x_{p}^{2}-x_{p+1}^{2}-\cdots-x_{d}^{2}$ and $x\cdot y=x_{1}y_{1}+\cdots+x_{p}y_{p}-x_{p+1}y_{p+1}-\cdots-x_{d}y_{d}$.

We have $d\psi(x)=dq(x)=2(x\cdot dx)$. Following Corollary 3.2 and Proposition 3.8, at each point $x$ of the canonical model, connection $\nabla$ is given by the following general formula:

[TABLE]

The parallel transport of a vector $v$ along a path $\gamma(t)$ is then characterized by the following differential equation:

[TABLE]

In this article, we call unit vector a vector $v$ such that $q(v)\in\{-1,1\}$, and we call orthonormal a family of unit vectors that are pairwise orthogonal.

6.1. The equipotential and radial foliations

The light cone is the set of $x\in E$ such that $q(x)=0$. The radial foliation is the set of straight half lines from [math] (excluded). Level sets of the potential $\psi$ form in any canonical model the equipotential foliation, whose leaves are quadric hypersurfaces, which are smooth, apart at $x=0$. The equipotential foliation is preserved by the action of the generalized orthogonal group $\operatorname{O}(p,d-p)$.

An arc not contained in the light cone is a leaf of the radial foliation if and only if at each point it is orthogonal to the equipotential foliation. Outside the light cone, the radial foliation and the equipotential foliation intersect transversely. The light cone is an equipotential leaf and is the union of $\{0\}$ and of a collection of rays.

6.2. Parallel transport along rays

In this Section, we consider the particular case of a parallel transport along a ray that does not belong to the light cone: $\gamma(t)=te_{r}$ where $t$ belongs to some interval $[t_{0},t_{1}]\subset\mathbb{R}^{*}_{+}$ and $e_{r}$ is a vector with $q(e_{r})=\pm 1$.

Any vector $v$ can be written in the form $v_{r}e_{r}+v_{\theta}e_{\theta}$ where $e_{\theta}$ is an orthoradial vector (parallel to the equipotential foliation at $\gamma(t_{0})$). The parallel transport of $v$ along a segment of a ray remains in the plane generated by $e_{r}$ and $e_{\theta}$ (see Corollary 3.6). We denote $v=v(t)=v_{r}e_{r}+v_{\theta}e_{\theta}$ the transported vector with $v_{r}=v_{r}(t)$ and $v_{\theta}=v_{\theta}(t)$.

If $v$ is a radial vector, then $v_{\theta}=0$ and Equation (6.2) simplifies to:

[TABLE]

The vector remains radial all along the path and the solutions are then $v_{\theta}(t)=0$ and $v_{r}(t)=k(\lambda+t^{2}q(e_{r}))^{2}=k(\lambda+q(x))^{2}$ with $x=\gamma(t)=te_{r}$, where $k=1/(\lambda+q(\gamma(t_{0})))^{2}\in{\mathbb{R}}$. In particular

[TABLE]

If $v$ is an orthoradial vector then $v_{r}=0$ and we have $x\cdot v=0$ and the equation simplifies to:

[TABLE]

The transported vector remains orthoradial all along the path and the solutions are $v_{r}(t)=0$ and $v_{\theta}(t)=k(\lambda+t^{2}q(e_{\theta}))=k(\lambda+q(x))$ where $k=1/\lambda+q(\gamma(t_{0}))$. In particular

[TABLE]

Note in particular, that the exponent of the $\psi(\gamma(t))$ factors is different for radial and orthoradial vectors.

Proposition 6.1**.**

Consider a ray $\cal R$ not contained in the light cone and let $e_{r}$ be a unit vector in $\cal R$. In an orthonormal basis where $e_{r}$ is the first vector, the parallel transport along a path $\gamma$ contained in $\cal R$ corresponds to a diagonal matrix:

[TABLE]

where $\Lambda=\frac{\psi(\gamma(t_{1}))}{\psi(\gamma(t_{0}))}$. The radial and each orthoradial direction are preserved. In particular, the tangent hyperplane to the equipotential foliation is preserved.

Parallel transport along rays in the light cone. The domain of a canonical model is the locus where $\lambda+q(x)\neq 0$. In particular, if $\lambda\neq 0$, a point $x$ in the model may belong to the light cone based on [math], i.e. the locus where $\psi=\lambda$. The computations performed above are still valid, and since $\psi$ is constant (and equal to $\lambda$) on the ray, they yield that $v(t)$ remains constant provided $v(0)$ is radial or orthoradial.

The only difference is that the radial and orthoradial subspaces no longer generate the whole vector space: actually the radial direction is now contained in the orthoradial hyperplane.

To treat the general case, let us write $\gamma(t)=te_{r}$ with $e_{r}\cdot e_{r}=q(e_{r})=0$, so $\dot{\gamma}=e_{r}$ and $\gamma\cdot\dot{\gamma}=0$. From Equation (6.2), for any initial vector $v(t_{0})$, the transport equation reads

[TABLE]

It follows that $v(t)=v(t_{0})+f(t)e_{r}$. Knowing this, the transport equation is equivalent to

[TABLE]

Hence

[TABLE]

By properties of parallel transport of connections, using a non-uniformly parameterized ray $\gamma(t)=f(t)e_{r}$ does not basically change the result. Here is a summary:

Corollary 6.2**.**

For any path $\gamma$ contained a radial leaf contained in the light cone:

• •

The parallel transport of a vector tangent to the light cone along $\gamma$ is trivial (constant).

• •

Writing $\gamma(t)=f(t)e_{r}$ with $q(e_{r})=0$, the parallel transport of a general vector $v$ along $\gamma$ satisfies

[TABLE]

6.3. Parallel transport along equipotential arcs

Outside the light cone, equipotential leaves are quadric hypersurfaces of $E$. We will not to compute the parallel transport along any path contained in such leaves, but focus on planar sections of these leaves. By plane we mean $2$-dimensional vector (or affine) subspaces of $E$.

Definition 6.3**.**

In any canonical model, we call equipotential arc a non-empty portion of the intersection of an equipotential leaf different from the light cone, with a plane $\mathcal{P}$ containing [math] and whose induced pseudo-Euclidean structure is nondegenerate.

Every point not in the light cone belongs to at least one equipotential arc, as follows from the following (slightly more general) lemma:

Lemma 6.4**.**

If $d\geq 2$, then for any $x\neq 0$ there exists a plane through $x$ and [math] on which $q$ is nondegenerate.

Proof.

If $q(x)\neq 0$, ${\mathbb{R}}x$ is transversal to its orthogonal, on which $q$ cannot vanish, for otherwise the scalar product would vanish too and hence any $y\in({\mathbb{R}}x)^{\perp}$ would be in the kernel of $q$, contradicting nondegenracy of $q$ on $E$; if $q(x)=0$ then since $x\notin\{0\}=\operatorname{Ker}q$, there exists $y\in E$ such that $x\cdot y\neq 0$. Then the determinant of the matrix of $q$ in basis $(x,y)$ is equal to $-(x\cdot y)^{2}\neq 0$. ∎

On an equipotential arc $\gamma$ of model $\cal S^{\nu}_{\lambda}(p,d-p)$ with $\nu\in\{-,+\}$ we have $q(\gamma)=\operatorname{cst}\neq 0$ and we let $r>0$ be such that151515Where $\nu=-$ is replaced by $-1$ and $\nu=+$ by $1$.

[TABLE]

Depending on whether the signature of the pseudo-Euclidean structure of the plane is $(2,0)$, $(0,2)$ or $(1,1)$, an equipotential $\gamma$ arc is parameterized by an usual angle or a hyperbolic angle $\theta$, characterized as follows: $|q(\dot{\gamma}(\theta))|=|q(\gamma(\theta))|=r^{2}$. Since $\gamma$ is an equipotential arc, the sign of $q(\dot{\gamma})$ is determined in advance: $q(\dot{\gamma}(\theta))=\epsilon\,q(\gamma(\theta))$ with $\epsilon=-1$ in signature $(1,1)$ and $\epsilon=1$ otherwise. Yet, this still allows for two opposite values of $\dot{\gamma}(\theta)$. To fix one, we choose any of the two possible orientations on the curve $\cal L$, where $\cal L$ is the component containing $\gamma$ of the equipotential leaf in $\cal P$. Then we impose $\dot{\gamma}(\theta)$ to be in the positive sense of this orientation.

By the nondegeneracy assumption, and because we are not on the light cone, there is a moving orthonormal frame $(e_{r},e_{\theta})$ of $\mathcal{P}$ so that

• •

$\gamma(\theta)=re_{r}$,

• •

$\dot{\gamma}(\theta)=re_{\theta}$.

As is customary for instance in physics, the symbol $r$ plays two different roles and $\theta$ too: as indices of $e$ here or $v$ below, they are purely symbolic and allow to tell one object from another. We have

• •

$\dot{e_{r}}=e_{\theta}$

• •

$\dot{e_{\theta}}=-\epsilon e_{r}$

where161616We could have taken the opposite convention for $\epsilon$ so as to have the simpler expression $\dot{e_{\theta}}=\epsilon e_{r}$ but our choice makes most other expressions simpler.

• •

$\epsilon=1$ in signatures $(2,0)$ and $(0,2)$;

• •

$\epsilon=-1$ in signature $(1,1)$.

We first observe the following fact. By the nondegeneracy assumption, $\cal P$ is transverse to its orthogonal space in $E^{p,d-p}$. For any vector $v$ that is orthogonal to $\mathcal{P}$, the parallel transport along $\gamma$ of $v$ is trivial. Indeed, vector $v$, position vector $\gamma$ and speed vector $\dot{\gamma}$ are orthogonal to each other. Equation 6.2 thus reduces to $\dot{v}=0$. In other words, the parallel transport along an equipotential arc contained in a plane $\mathcal{P}$ preserves any vector orthogonal to $\mathcal{P}$.

Now we consider a vector $v$ of the form $v_{r}e_{r}+v_{\theta}e_{\theta}$. The parallel transport of $v$ along $\gamma$ always remains in $\mathcal{P}$ (by Corollary 3.6). Equation (6.2) for the parallel transport of $v$ along $\gamma$ reduces to:

[TABLE]

Recall that $q(\gamma)$ is constant. Since $\dot{v}=(\dot{v_{r}}-\epsilon v_{\theta})e_{r}+(v_{r}+\dot{v_{\theta}})e_{\theta}$, we obtain a system of two differential equations:

• •

$\dot{v_{r}}-\epsilon v_{\theta}=0$;

• •

$v_{r}+\dot{v_{\theta}}=\frac{2q(\gamma)}{\lambda+q(\gamma)}v_{r}$.

The system is then equivalent to:

• •

$\epsilon v_{\theta}=\dot{v_{r}}$;

• •

$\ddot{v_{r}}=\epsilon\frac{q(\gamma)-\lambda}{q(\gamma)+\lambda}v_{r}$.

One recognizes the harmonic oscillator equation with position variable $v_{r}$ and speed variable $\epsilon v_{\theta}$.

For any equipotential leaf (where $q$ is constant), we introduce the characteristic frequency

[TABLE]

with $\lambda^{\prime}=\lambda/q(\gamma)=\nu\lambda/r^{2}$. Recall that the connection is not defined on the locus where $q+\lambda$ vanishes, so $q(\gamma)+\lambda\neq 0$ in our case.

The quantity $\omega$, defined up to a sign, is either a real number or a purely imaginary number (we can fix $\omega$ by imposing that $\operatorname{Re}\omega$ and $\operatorname{Im}\omega$ be non-negative):

• •

if $\lambda=0$, then $\omega=1$ in signatures $(2,0)$ or $(0,2)$ and $\omega=i$ in signature $(1,1)$;

• •

if $q(\gamma)=\lambda$ then $\omega=0$;

• •

if $|q(\gamma)|>|\lambda|$, then $\omega$ is a real number in signatures $(2,0)$ or $(0,2)$, purely imaginary in signature $(1,1)$;

• •

if $|q(\gamma)|<|\lambda|$, then $\omega$ is a purely imaginary number in signatures $(2,0)$ or $(0,2)$, real in signature $(1,1)$;

In the case $\omega\in\mathbb{R}$, solving the harmonic oscillator equation provides the following solution:

[TABLE]

If $\omega$ is purely imaginary, we consider $\alpha=-i\omega$. Then $\alpha\in{\mathbb{R}}$ and we obtain the following solution:

[TABLE]

Actually the two matrices coincide in all cases using complex versions of the trigonometric and hyperbolic functions. Note also that choosing the opposite square root for $\omega$ does not change the matrices given by the formulae above, as expected regarding the fact that the parallel transport does not depend on the choice of $\omega$.

Do not forget that the matrices above are expressed in a polar coordinate system. Unless we are in signature $(2,0)$ or $(0,2)$ and perform a full turn around the (circular) leaf, for the vector space structure of $\cal P$, this matrix is expressed in bases in the domain and range that are different bases of the plane $\cal P$. To recover its expression in a single basis of $\cal P$, one has to multiply it on each side with appropriate transition matrices.

Remarkably, if $\omega=0$ then the coordinates $v_{r}$ and $v_{\theta}$ are constant, i.e. $v$ is constant in the moving frame, but as we just saw, $v$ is not constant in a basis of $E$. An equipotential leaf with $\omega=0$ exists on the models $\cal S^{+}_{1}(2,0)$, $\cal S^{-}_{-1}(0,2)$, $\cal S^{+}_{1}(1,1)$ and $\cal S^{-}_{-1}(1,1)$ but not on the 8 others canonical models in dimension 2. In signatures $(2,0)$ and $(0,2)$ these are the models whose domain are the whole plane. In signature $(1,1)$ these are the models that contain [math].

Performing a full turn along a circular leaf. In the case of signature $(2,0)$, by taking $t=2\pi$, we obtain the holonomy matrix corresponding to a loop, more precisely a full turn along the circle of Euclidean radius $r$.

Remarkably, when $\lambda=0$, this holonomy matrix is independent of the leaf and is

[TABLE]

Numerical values are $\cosh 2\pi\approx 267.7467615$ and $\sinh 2\pi\approx 267.7448940$.

Still in the case $\lambda=0$ in signature $(2,0)$, note that the parallel transport along closed or non-closed arcs of circles are isometries for the pseudo-Riemannian structure $v_{r}^{2}-v_{\theta}^{2}=(dr)^{2}-r^{2}(d\theta)^{2}$, whereas the Riemannian structure induced by the Euclidean norm has expression $v_{r}^{2}+v_{\theta}^{2}=(dr)^{2}+r^{2}(d\theta)^{2}$.

In the case $\lambda=-1$ and $\nu=-1$, for which the domain of $\cal S$ is the Euclidean unit disk, if we take the circle of Euclidean radius $r$, we get the holonomy matrix

[TABLE]

with

[TABLE]

When $r\longrightarrow 0$ we get the expansion

[TABLE]

In other words, a transported vector asymptotically turns by $-2\pi r^{2}$. This is coherent with the value $-2$, at $x=0$, of the Gaussian curvature of $\nabla$ relative to the flat metric $g$, which we found is equal to $K/\psi=2/\psi$. Here $\psi(0)=-1$ and the Euclidean area the disk of Euclidean radius $r$ is $\pi r^{2}$.

Model $\cal S^{\pm}_{0}(1,1)$. For $\lambda=0$ in signature $(1,1)$ the holonomy matrices for paths along leaves (different from the light cone) are the matrices

[TABLE]

where $\theta$ measures the hyperbolic angle difference along the leaf where the sign convention is with respect to the initially chosen orientation on the (one dimensional) leaf $\cal L$. These are isometries for the riemannian structure $v_{r}^{2}+v_{\theta}^{2}=(dr)^{2}+r^{2}(d\theta)^{2}$, whereas the pseudo-Riemannian structure induced by the pseudo-Euclidean norm has expression $\nu\times(v_{r}^{2}-v_{\theta}^{2})=\nu\times((dr)^{2}-r^{2}(d\theta)^{2})$.

6.4. Local holonomy

The connection of the canonical models are incompressible, so the holonomy of contractible loops is contained in the group $\operatorname{SL}(d,\mathbb{R})$. We prove in this section that the local holonomy groups $\operatorname{Hol}(x,V)$ (generated by the holonomy of loops contained in a neighbourhood $V$ of $x$) always coincide with $\operatorname{SL}(d,\mathbb{R})$

Proposition 6.5**.**

In any canonical model $\cal S=\mathcal{S}_{\lambda}^{\pm}(1,1)$ or $\cal S=\mathcal{S}_{\lambda}^{\pm}(2,0)$, for any open subset $V$ of $\cal S$ and any $x\in V$, the holonomy group $\operatorname{Hol}(x,V)$ of the connection is equal to $\operatorname{SL}(2,\mathbb{R})$.

Proof.

We already know that the holonomy locally preserves some volume form hence is contained in $\operatorname{SL}(2,\mathbb{R})$. Parallel transport locally conjugates holonomy groups in $V$. It is hence enough to consider the case where $x$ is not on the light cone (whose interior is empty). By Theorem 4.2 of [KN63], $\operatorname{Hol}(x,V)$ is a Lie subgroup of $SL(2,\mathbb{R})$. Its Lie algebra $\cal L$ contains the infinitesimal holonomies $M(i,j)$ mentioned in Section 2, whose matrices satisfy $M(i,j)_{kl}=-R_{ijk}^{l}$. By Corollary 4.7, $M(1,2)$ has matrix $\frac{K}{\psi}J_{\sigma}$ with $J_{\sigma}=\begin{pmatrix}0&-\epsilon_{j}\\ \epsilon_{i}&0\end{pmatrix}$. In particular, $J_{\sigma}\in\cal L$. Consider now the parallel transport along the ray through $x$ and its expression in an adapted basis $(e_{r},e_{\theta})$. This new basis is also orthonormal and the change of bases turns $J_{\sigma}$ into $PJ_{\sigma}P^{-1}$ which equals either $J_{\sigma}$ or $-J_{\sigma}$. Let $\tau>1$ be small. According to 6.2, the parallel transport from $\tau x$ to $x$ has, in the basis $(e_{r},e_{\theta})$, matrix $D=\operatorname{Diag}(\Lambda^{2},\Lambda)$ where $\Lambda=\frac{\psi(x)}{\psi(\tau x)}$. Note that $\Lambda$ is close to $1$, but different. If we transport the infinitesimal holonomy element $J_{\sigma}$ from $\tau x$ to $x$ we obtain $DJ_{\sigma}D^{-1}=\begin{pmatrix}0&-\Lambda\epsilon_{j}\\ \Lambda^{-1}\epsilon_{i}&0\end{pmatrix}$. Differentiating as $\tau$ tends to $1$ we get that $K_{\sigma}\in P\cal LP^{-1}$ where $K_{\sigma}=\begin{pmatrix}0&\epsilon_{j}\\ \epsilon_{i}&0\end{pmatrix}$. One computes $[J_{\sigma},K_{\sigma}]=-2\epsilon_{i}\epsilon_{j}L_{\sigma}$ with $L_{\sigma}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$, hence $L_{\sigma}\in P\cal LP^{-1}$. This proves the proposition, since $(J_{\sigma},K_{\sigma},L_{\sigma})$ is a basis of the Lie algebra of $\operatorname{SL}(2,{\mathbb{R}})$. ∎

Theorem 6.6**.**

In any canonical model $\mathcal{S}_{\lambda}^{\pm}(p,d-p)$, for any open subset $V$ and any $x\in V$, the holonomy group $\operatorname{Hol}(x,V)$ of the connection is $\operatorname{SL}(d,\mathbb{R})$.

Proof.

We already know that the holonomy locally preserves some volume form hence is contained in $\operatorname{SL}(d,\mathbb{R})$. Again, it is enough to compute the holonomy group in the neighborhood of a point $x$ that does not belong to the light cone. Then the vector $x-0$ is disjoint from its orthogonal hypersurface $H$ (the tangent space at $x$ to the foliation is $x+H$). The form $q$ is nondegenerate on $H$ (otherwise it would be degenerate on ${\mathbb{R}}^{d}$) so we can choose an orthonormal basis $e_{2},\dots,e_{d}$ of $H$, and complete it into an orthonormal basis $e_{1},\dots,e_{d}$ of ${\mathbb{R}}^{d}$ where $e_{1}$ is a radial vector. We denote by $\mathcal{L}$ the Lie algebra that is tangent to $\operatorname{Hol}(x,V)$. We consider the holonomy of loops belonging to a plane $\cal P$ through $x$ and directed by the vector plane $P$ generated by $e_{1}$ and some other element $e_{i}$ of the basis. Following Sections 6.2 and 6.3, these holonomies are the identity on $P^{\perp}$.

On $\cal P$, the parallel transport is equivalent to a nondegenerate two-dimensional model. From this and Proposition 6.5, we get that for any $2\leq i\leq d$, $\mathcal{L}$ contains a copy of $\mathfrak{sl}(2,{\mathbb{R}})$. More precisely, the following base elements of $\mathfrak{sl}(2,{\mathbb{R}})$: $\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$, $\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$, $\begin{pmatrix}0&0\\ 1&0\end{pmatrix}$ correspond respectively in $\mathfrak{sl}(d,{\mathbb{R}})$ to $e_{1}\otimes e_{1}^{\ast}-e_{i}\otimes e_{i}^{\ast}$, $e_{1}\otimes e_{i}^{\ast}$ and $e_{i}\otimes e_{1}^{\ast}$.

For $i\neq j$, the Lie bracket of $e_{j}\otimes e_{1}^{\ast}$ and $e_{1}\otimes e_{i}^{\ast}$ equals $e_{i}\otimes e_{j}^{\ast}$. It follows that the Lie algebra $\mathcal{L}$ contains the basis $e_{1}\otimes e_{1}^{\ast}-e_{i}\otimes e_{i}^{\ast}$ ($i\neq 1$), $e_{i}\otimes e_{j}^{\ast}$ ($i\neq j$), of $\mathfrak{sl}(d,\mathbb{R})$. The local holonomy group of the connection thus coincides with $\operatorname{SL}(d,\mathbb{R})$. ∎

7. A tale of three metrics (and five volume forms)

Let us recall that a conformal structure or conformal class for signature $s$ on a manifold is a choice of on each tangent space of a pseudo-Euclidean space structure up to rescaling, of this signature. In other words, a quadratic form of signature $s$, up to multiplication by a constant. On the manifold, this means the choice of a pseudo-Riemannian structure of signature $s$ up to multiplication by a positive function.

Any canonical model $\cal S^{\nu}_{\lambda}(p,q)$, $\nu\in\{-,+\}$ with its connection $\nabla$ takes place within a pseudo-Euclidean space $E^{p,d-p}$, endowed with a quadratic form $q(x)=\sum\epsilon_{i}x_{i}^{2}$ and the associated scalar product $(x,y)\in E^{2}\mapsto x\cdot y\in{\mathbb{R}}$ and the associated flat pseudo-Riemannian tensor

[TABLE]

characterized also by $g_{ij}=\delta_{ij}\epsilon_{i}$ or equivalently as the symmetric tensor such that

[TABLE]

However, $g$ is not preserved by the connection $\nabla$.

On the opposite, $\nabla$ has an unambiguous Ricci tensor

[TABLE]

(see Corollary 4.5, where we have $K=2$). If $\nu<0$ then $\psi<0$, so the signature of the Ricci tensoir is $(d-p,p)$ (the opposite signature). If $\nu>0$ then $\psi>0$, so the signature of the Ricci tensor is $(p,d-p)$ (the same as for $g$).

Remark 7.1*.*

In [Kob08] has been given many examples of incompressible connections with positive definite or negative definite Ricci tensors. Our canonical models provide another family of examples when $(p,d-p)=(d,0)$ or $(0,d)$.

As we will see in Section 7.1, we can define another pseudo-Riemannian metric $h$ with remarkable properties: it is in the conformal class of $g$, the geodesics of $\nabla$ have constant speed with respect to $h$, and its expression is

[TABLE]

where $\psi$ is the potential.

In the end, we find in the same conformal class three metrics:

1. (1)

the flat pseudo-Euclidean metric $g$; 2. (2)

the Ricci tensor if $\psi>0$ (or its opposite if $\psi<0$) $\mathrm{Ric}=\frac{2d-2}{\psi}g$; 3. (3)

the isochrone metric $h=\frac{g}{\psi^{4}}$.

7.1. Isochrone metric

For any projectively flat connection, an explicit parameterization of geodesics can be given in terms of the potential $\psi$ (see Section 3.6). In the case of stiff connections, we can encompass the parameterization data into a pseudo-Riemannian metric conformally equivalent to the underlying pseudo-Euclidean structure, as we explain below.

Definition 7.2**.**

A metric and a connection are isochrone to each other if the geodesics of the connection have constant speed with respect to the metrics.

Note that geodesics for the metric do not have to be geodesics for the connection, nor conversely.

Proposition 7.3**.**

In any canonical model, $\nabla$ is isochrone to

[TABLE]

where $\alpha>0$ and $g$ is the usual pseudo-Euclidean metric. Moreover, these are locally the only isochrone metrics for $\nabla$ in the same conformal class as $g$ (by this we mean metrics of the form $\rho(x)g$ where $\rho$ is a positive function).

Proof.

The fact that $h$ is isochrone to $\nabla$ is proved in Section 3.6. Conversely any two isochrone and $g$-conformal metric of the form $\rho g$ and $\tilde{\rho}g$ must locally have a constant ratio $\alpha=\tilde{\rho}/\rho$ on lines whose direction is not in the light cone. Locally any point close to $x$ can be reached by following one or two straight segments whose directions are not in the light cone. It follows that $\tilde{\rho}/\rho$ is locally constant. ∎

We make three remarks.

If the geodesic direction $\dot{\gamma}$ is in the light cone of $q$, i.e. if $q(\dot{\gamma})=0$, then the speed w.r.t $g$, hence to $h$, is automatically [math] along $\gamma$, hence the isochrone property does not bring any information for these specific geodesics. Yet, the geodesic is isochrone to $\tilde{g}/\psi^{4}$ for any Euclidean metric $\tilde{g}$ on its supporting line, see Section 3.6.

For any $\alpha$, the isochrone metric $h$ is not invariant by the parallel transport of the connection.

Consider a canonical model and a choice of isochrone metric $h=\frac{\alpha^{2}}{\psi^{4}}g$. It would be tempting to define a distance adapted to $\nabla$ as follows: given two points $a$ and $b$ let $d(a,b)=$ the distance between $a$ and $b$ w.r.t. the metric $h$ restricted to $[a,b]$. However there are three problems:

• •

one has to assume that the straight segment $[a,b]$ is contained in the domain of the model, so not necessarily every pair of points would be joinable, depending on the model;

• •

the triangle inequality is not necessarily satisfied;171717See Section 7.6 for counterexamples.

• •

in mixed signatures, the distance would be positive, zero or imaginary, depending on the sign of $q(b-a)$.

7.2. More about the isochrone metric

In this section we investigate properties of the isochrone metric per se, i.e. without having in mind applications to the study of $\nabla$.

Consider the isochrone metric

[TABLE]

i.e. we choose $\alpha=1$ to fix ideas. First we would like to compute the scalar curvature $S(h)$ of $h$. We recall that this is a $(0,0)$-tensor (a.k.a. a function), i.e. it is independent of the chart. On the other hand, if one chooses another isochrone metric by multiplying $h$ by $\alpha^{2}$, this multiplies $S$ by $\alpha^{-2}$.

To compute the scalar curvature we use the formula (see [Bes87], Chapter 1, Section J, Theorem 1.159)

[TABLE]

where $g$ is a general pseudo-Riemannian metric, $\Delta_{g}$ is the pseudo-Laplacian associated to $g$ and the musical isomorphism is used to define $g(df,df)$. In the case of our flat $g=\sum_{i}\epsilon_{i}dx_{i}\otimes dx_{i}$, we have $S(g)=0$ and $\Delta_{g}=\sum_{i}\epsilon_{i}\frac{\partial^{2}}{(\partial x_{i})^{2}}$. Then letting $f=-2\log|\psi|$ and remembering that $\psi=q+\lambda$ we get:

[TABLE]

Remarkably, $S(h)$ vanishes on the boundary of the canonical model.

In dimension $d\geq 3$, the sign of $S(h)$ will change on the set of equation $q(x)=\frac{d}{d-2}\lambda$, which is non-empty in the domain $U$ of $\cal S^{\nu}_{\lambda}(p,d-p)$ if and only if: $\lambda\neq 0$, $\lambda$ and $\nu$ have the same sign and there is at least one $\epsilon_{i}$ of this sign. In this case it is a smooth equipotential. Moreover, $U$ contains [math] and $S(h)$ has the sign of $\lambda$ at $x=0$. If one replaces $\nu$ by $-\nu$, assuming that the domain of $\cal S^{-\nu}_{\lambda}(p,d-p)$ is non-empty, then the sign of $S(h)$ is constant and equal to the sign of $\lambda$.

In dimension $d=2$, we get the simpler expression

[TABLE]

so the sign of the scalar curvature of $h$ is constant and the same as the sign of $\lambda$. If moreover $\lambda=0$ then $S(h)=0$, which has the remarkable implication that $h$ is flat. Actually, a flattening map, i.e. a (local) diffeomorphism sending $h$ to $g$, can be explicitly determined:

• •

In signatures $(2,0)$ or $(0,2)$, identifying $E$ with ${\mathbb{C}}$, this is the non-injective map $z\mapsto 1/3z^{-3}$, whose expression in polar coordinates is $(r,\theta)\mapsto(1/3r^{3},-3\theta)$.

• •

In signature $(1,1)$ this is the diffeomorphism from $U$ to itself whose expression in polar hyperbolic coordinates is the same: $(r,\theta)\mapsto(1/3r^{3},-3\theta)$. This map has a rational expression in terms of the coordinates $x_{1},x_{2}$, that we omit here.

One may want to compare the sign of $S(h)$, which is the sign of $\lambda$ according to Remark 4.18 with the sign of the relative scalar curvature $S(\nabla,g)$, which is $\nu$ according to Remark 4.9. Hence they do not necessarily match. We found interesting to present the comparison of the different possible signs in Figure 3. Note that they agree if and only if the domain contains [math].

Let us repeat here the interpretation of the sign of scalar curvature in dimension $2$ that we gave at the end of Section 4.2. Under the infinitesimal holonomy associated to small injective loops, vectors undergo an infinitesimal isometry. The latter displaces a vecto $v$ along a direction which is tangent to the unit “sphere” $q(v)=\operatorname{cst}$, in a direction that depends on the product of the sign of $q(v)$, the sign of the scalar curvature and the orientation of the loop: they move in the same direction as the loop if this product is positive, in the other direction otherwise.

Coming back to any dimension $d\geq 2$, denote $\psi_{\lambda}=q(x)+\lambda$. If $\lambda\neq 0$, we have $h=g/\psi_{\lambda}^{4}$ and $1/\psi_{\lambda}^{4}=1/(q(x)+\lambda)^{4}\sim 1/q(x)^{4}=1/\psi_{0}^{4}$ when $|q(x)|\to+\infty$. Hence in some way, if the model is unbounded, the isochrone metrics for $\lambda\neq 0$ are tangent to the isochrone metrics for $\lambda=0$ (one can reach infinity by a path of finite length for $h$: it is enough to take a straight line whose direction $v$ satisfies $q(v)\neq 0$). However one also notices that for $\lambda\neq 0$, $|S(h)(x)|\to+\infty$ when $|q(x)|\to+\infty$ whereas $S(h)=0$ for $\lambda=0$, so the tangency is not that good.

In any model, geodesics of the isochrone metrics $h=\alpha^{2}g/\psi^{4}$ depend on $\alpha>0$ as parameterized curves but their support is independent of $\alpha$. By pure curiosity, we drew geodesics of the isochrone metrics of a few dimension $2$ models in Figure 4.

7.3. Forms and curvature

We saw in Proposition 3.8 that the volume forms preserved by the connection of a canonical model are

[TABLE]

with $\beta\in\mathbb{R}^{*}$. We make here two remarks.

Canonical model $\mathcal{S}_{1}^{+}(d,0)$, i.e. the one whose domain is all ${\mathbb{R}}^{d}$, has a finite total volume for any of these forms. We can check that for any other canonical model, the total volume is infinite.

The preserved volume forms $\omega$ are different from the volume form induced by the flat metric:

[TABLE]

and from the volume forms induced by the isochrone metrics $h=(\alpha^{2}/\psi^{4})g$:

[TABLE]

for some $\beta>0$.

In dimension $d=2$, we would like to push a little bit further the analogy with the Levi-Civita connection of a Riemannian metric $\rho$. More precisely, in dimension $2$ a particular case of the Gauss-Bonnet theorem can be stated in terms of a curvature form

[TABLE]

where $G$ denotes the Gaussian curvature and $\operatorname{vol}_{\rho}$ the volume form canonically associated to $\rho$: for any injective loop anticlockwise encompassing an oriented and simply connected domain $U$, the holonomy of the loop is a rotation whose angle, as measured per the orientation, is given by the integral of $\kappa_{\rho}$ on $U$ with respect to its orientation.

Such a strong statement cannot hold in our case since the local holonomy groups are all of $\operatorname{SL}(2,{\mathbb{R}})$ (Proposition 6.5). However, first-order infinitesimal holonomies are rotations, and we computed in Equation 4.1 that $M(1,2)=\frac{2}{\psi}\cal J(1,2)$ (recall that we fixed the constant $K=2$) where $M(1,2)$ is the first-order infinitesimal holonomy associated to basis $(e_{1},e_{2})$ of the global chart ${\mathbb{R}}^{2}$ and $\cal J(1,2)$ can be considered as a canonical infinitesimal rotation in the chart. In view of the geometric interpretation of infinitesimal holonomies, it makes sense to define

[TABLE]

and call it curvature form too, although it only satisfies Gauss-Bonet at the first order (when the loop size tends to [math]). We defined after Remark 4.9 the Gaussian curvature $G(\nabla,g)$ of $\nabla$ relative to $g$. Note that $\kappa$ is also the product of this quantity with $\operatorname{vol}_{g}$, i.e. $\kappa_{\nabla}=G(\nabla,g)\operatorname{vol}_{g}$. It would be tempting to define the relative curvature as the quotient of the curvature form $\kappa$ by the invariant form $\omega$. One problem is that the latter is only unique up to multiplication by a constant:

[TABLE]

for any $\beta\neq 0$. Anyway, one immediately gets that the relative curvature is $2\psi^{4}/\beta$. Remarkably, it vanishes on the boundary of the domain of the canonical model.

The curvature form of the isochrone metric is

[TABLE]

It is thus different from the curvature form $\omega$ that we defined in equation (7.1).

It is interesting to watch the following list:

[TABLE]

7.4. Parameterization of geodesics

Here we explain how to solve explicitly the geodesic equation in the canonical models. Recall that all geodesics of $\nabla$ are straight lines of the pseudo-Euclidean space in which the model sits. We can either use the isochrone metric mentioned in previous sections or use the results of Section 3.6. We choose the second approach, which has the advantage to also work when the derivative vector of the geodesic is in the light cone.

Let $t\mapsto\gamma(t)$ be a geodesic and $L$ be the affine line in $E$ containing the image of $\gamma$. By proposition 3.16, the restriction of $\psi$ to $F$ is a potential for the restriction of $\nabla$ to the affine subspace $L$. Choose any affine bijection $\phi:{\mathbb{R}}\to L$, $s\mapsto x_{0}+se$, where $e\in\overrightarrow{L}$ is not zero (it makes sense to take $x_{0}=\gamma(0)$ and $e=\dot{\gamma}(0)$ but the discussion also works for other choices). We can consider on ${\mathbb{R}}$ the connection $\phi^{*}(\nabla)$, one of whose potentials is given by

[TABLE]

The index $s$ is purely symbolic and denotes the variable in which we work. The geodesic $\gamma$ in this new coordinate system becomes a geodesic for $\phi^{*}(\nabla)$. The function $\psi_{s}$ develops as $\psi_{s}(s)=s^{2}q(e)+2(x_{0}\cdot e)s+q(x_{0})+\lambda$.

The potential $\psi_{s}$ is a polynomial of degree at most 2 over ${\mathbb{R}}$ and we will consider different cases according to its actual degree. Note that $\psi_{s}$ cannot be the zero polynomial, for otherwise $\psi_{s}$ would vanish on $L$ but it cannot vanish at the initial point $\gamma(0)$ of the geodesic. Note that the roots of $\psi_{s}$ are in correspondence with the intersection of $L$ with the equipotential $\psi=0$.

By a further affine change of variable $s\mapsto y$ we can reduce to studying the geodesic equation of the connection generated by the following five potentials:

1. (1)

$\psi_{y}=1$ 2. (2)

$\psi_{y}=y$ 3. (3)

$\psi_{y}=y^{2}-1$ 4. (4)

$\psi_{y}=y^{2}$ 5. (5)

$\psi_{y}=y^{2}+1$

In the first case, no change of variable is required. In the second case, only a translation is needed. This classifies geodesics into five types. One may subdivide (3) into further sub-cases: $|y|<1$ and $|y|>1$.

Since $\psi_{y}$ vanishes exactly at parameters corresponding to the intersections of $L$ and the equipotential $\psi=0$, we know that any connected component of the open subset of ${\mathbb{R}}$ where $\psi_{y}$ does not vanish corresponds to an interval of $L$ contained in one of the canonical model. Two different such intervals may be contained in the same model, depending on the situation.

Let us denote $\nabla^{y}$ the connection in the $y$ coordinate: it is generated by $\psi_{y}$ and in our $1$-dimensional setting, Section 3.6 teaches us that the antiderivative $F$ of any constant multiple of $1/\psi_{y}^{2}$ trivializes the connection.

Case 1: $\Psi_{y}=1$. Then $\nabla_{y}$ is already trivial (is the canonical connection in ${\mathbb{R}}$) and the geodesic $t\mapsto y(t)$ is an affine function of $t$, defined for all times $t\in{\mathbb{R}}$:

[TABLE]

for some $\alpha\in{\mathbb{R}}^{*}$ and $\beta\in{\mathbb{R}}$. It follows that in $E$, $\gamma$ is defined for all $t\in{\mathbb{R}}$ and is an affine parameterization of $L$. Case 1 occurs if and only if $L$ does not intersect the equipotential $\psi=0$. Equivalently when $q(e)=0$ and $(x_{0}\cdot e)=0$.

Case 2: $\Psi_{y}=y$. Then the geodesic of $\nabla^{y}$ is supported in either $(-\infty,0)$ or $(0,\infty)$ and since $-1/y$ is an antiderivative of $1/y^{2}$, the change of variable $y\mapsto 1/y$ trivializes the connection (see also Proposition 3.15). This implies that

[TABLE]

for some $\alpha\in{\mathbb{R}}^{*}$ and $\beta\in{\mathbb{R}}$. It follows that in $E$, the geodesic $\gamma$ is defined on a semi-infinite interval of times, that it has the form $\gamma(t)=x_{\infty}+\frac{\alpha}{(t/t_{0})-1}e$ where $x_{\infty}$ is the unique intersection of $L$ with the equipotential $\psi=0$ and $\alpha$ and $t_{0}$ are determined by the initial conditions. The geodesic tends in infinite time to $x_{\infty}$ with Euclidean distance $\sim\operatorname{cst}/|t|$ and it tends to $\infty$ in finite time, as $t\longrightarrow t_{0}$, with a growth of order $\operatorname{cst}/|t-t_{0}|$. Case 2 occurs if and only if $q(e)=0$ and $(x_{0}\cdot e)\neq 0$. Equivalently when $L$ intersects the equipotential $\psi=0$ at a unique point counted with multiplicity, i.e. this point is different from the origin and the intersection is transverse.

We are in cases 3, 4 or 5 if and only if $q(e)\neq 0$, and we can distinguish them for instance by the number of intersections of $L$ with the equipotential $\psi=0$. In these three cases we have $\psi_{y}=y^{2}+\lambda^{\prime}$ for some $\lambda^{\prime}\in\{-1,0,1\}$. Solving the geodesic equation then amounts to determining an antiderivative $F=\int dy/\psi_{y}^{2}$ and solving

[TABLE]

where $\alpha\in{\mathbb{R}}^{*}$ and $\beta\in{\mathbb{R}}$ depend on the initial conditions. We can choose:

• •

for $\lambda^{\prime}=1$, $\displaystyle 2F(y)=F_{1}(y):=\frac{y}{1+y^{2}}+\arctan y$;

• •

for $\lambda^{\prime}=0$, $\displaystyle 3F(y)=F_{0}(y):=-\frac{1}{y^{3}}$;

• •

for $\lambda^{\prime}=-1$, $\displaystyle 4F(y)=F_{-1}(y):=-\frac{1}{y-1}-\frac{1}{y+1}+\ln\left|\frac{y+1}{y-1}\right|$.

The graph of each of these three functions is shown in figures 5. Parameterizing the geodesic requires inverting $F$, which can be done numerically but seems hard or impossible to do symbolically, except in the case $\lambda^{\prime}=0$.

Still in cases 3, 4 and 5, when $y\longrightarrow\pm\infty$ we have $1/\psi_{y}^{2}\sim 1/y^{4}$. It follows that the function $F$ has necessarily limits at $+\infty$ and $-\infty$. It also follows that $F(y)-F(s\infty)\sim\frac{-1}{3y^{3}}$ as $y\longrightarrow s\infty$, where $s=\pm 1$. In particular, any geodesic that tends to infinity along $L$ will do so in finite time with a rate $\sim\kappa/|t-t_{0}|^{1/3}$ for some $\kappa>0$.

Case 3: $\Psi_{y}=y^{2}-1$. We distinguish whether the initial value $y(0)$ is between the roots of $y^{2}-1$ or outside.

In the first case, in $E$ the geodesic $\gamma$ is defined for all times $t\in{\mathbb{R}}$ and tends in the future and in the past to the two intersections of $L$ with the equipotential $\psi=0$, which is contained in the boundary of the domain of the canonical model. From the expression of $F_{-1}$ one gets that the distance181818for the metric associated to $g$ or any smooth metric on the vector line ${\mathbb{R}}e$ to the limit point is equivalent to $\kappa/|t|$ for some $\kappa>0$.

In the second case the geodesic tends to infinity in finite time in one direction and to the equipotential $\psi=0$ in infinite time in the other direction, with a distance proportional to $\kappa/|t|$ for some $\kappa>0$.

Case 4: $\Psi_{y}=y^{2}$. It is easy to compute $F_{0}^{-1}$ and we get that

[TABLE]

where $A\in{\mathbb{R}}^{*}$ and $t_{0}\in{\mathbb{R}}$ are constants determined by initial conditions of the geodesic. In $E$, the geodesic $\gamma$ reaches infinity in finite time, at $t=t_{0}$, and reaches in infinite time either the origin or the point where $L$ is tangent to the equipotential $\psi=0$. The above expression of $y$ gives the rate a which these convergences occur.

Case 5: $\Psi_{y}=y^{2}+1$. In $E$ the geodesic $\gamma$ reaches infinity in finite time in the future and in the past and goes through every point of $L$. In particular, in the $y$ coordinate, the formula for $F_{1}$ shows that a geodesic that starts from $y(0)=0$ with a derivative $\dot{y}(0)=1$ is defined from $t=-\pi/4$ to $t=\pi/4$.

Note: as a matter of comparison, the geodesics in the Beltrami-Cayley-Klein model, and in the Beltrami-Poincaré model, reach the boundary of the circle also in infinite time. However in both cases, the Euclidean distance to the limit point is exponentially small with respect to time, which is much faster than the rates $\kappa/|t|$ and $\kappa/|t|^{1/3}$ for geodesics of $\nabla$ tending to the boundary of a canonical model.

7.5. Restriction/Extension

It is to be noted that for any affine subspace $F$ of $E$, the potential $\psi$ is a polynomial of $\operatorname{dim}F$ variables and degree at most $2$. Conversely for any affine space $F$, and real polynomial of $P$ degree at most $2$ on $F$, it is an exercise in algebra to prove that there exists a nondegenerate pseudo-Riemannian space $(E,q)$ of bigger dimension, and an embedding $F\to E$, such that the quadratic form $q$ restricted to $F$ equals $P$. It is even possible to choose a single $E$ such that all dimension $d$ spaces $F$ with their $P$ can be embedded this way: $E=E^{d,d}$ does the job.

It follows in particular that the curvature-free projectively flattened connections, which we recall are images by a projective transformation of the canonical connection, and are given by $\psi(x)=\Lambda(x)$ where $\Lambda$ is a non-zero affine map, are all well-chosen restrictions of well-chosen canonical models.

7.6. Example

Consider model $\cal S^{-}_{-1}(2,0)$, defined in the unit disk $D$ of the classical Euclidean plane ${\mathbb{R}}^{2}$. We saw that the geodesics of the associated connection $\nabla$, which are portions of straight lines in ${\mathbb{R}}^{2}$, have constant speed with respect to the isochrone metric $h=g/\psi^{4}$ where $g=dx_{1}^{2}+dx_{2}^{2}$ and $\psi(x)=x_{1}^{2}+x_{2}^{2}-1$. Here we use Section 7.4 to detail steps to compute the time $T$ it takes to go from one point $a\in D$ to another $b$ at speed $1$.

First, by definition

[TABLE]

Consider the straight line $F$ of ${\mathbb{R}}^{2}$ going through $a$ and $b$. We are necessarily in case 3 of Section 7.4. The affine parameterization $y\in{\mathbb{R}}\to F$ leading to $\psi$ being proportional to $y^{2}-1$ is such that $-1$ and $1$ map to the intersection $u$ and $v$ of $F$ with $\partial D$. Let $\ell=\|u-v\|/2$ be half of the Euclidean length of $F\cap D$, so that $\|dx\|=\ell|dy|$. Points $a$ and $b$ correspond to two parameters $y_{a}$ and $y_{b}$ in $(-1,1)$, and

[TABLE]

Now $\psi\left(\frac{(1-y)u+(y+1)v}{2}\right)=\ell^{2}(y^{2}-1)$ hence

[TABLE]

for the map $F_{-1}$ described before Case 3 of Section 7.4.

It would be tempting to consider this time as a notion of distance between $x$ and $y$, however, it does not satisfy the triangle inequality as we show below on an example.

We consider $s\in(0,1)$ and the points $a=(s,0)$, $b=(0,s)$ and $o=(0,0)$. Then

[TABLE]

A numerical analysis shows that $T(a,b)\leq T(o,a)+T(o,b)$ if and only if $s\leq s_{0}\approx 0.687$. For instance for $s=0.9$, we have $T(a,b)\approx 8.18$ whereas $T(o,a)+T(o,b)\approx 6.21$. By taking the angle between $oa$ and $ob$ smaller, we can worsen the situation (this is coherent with the geodesics of $h$ being far from being straight chords of $D$, see Figure 4).

8. Global geometry of stiff connections on manifolds

In this section, we investigate global aspects of (similarity) pseudo-Euclidean manifolds endowed with a stiff connection. We are interested in classifying such structures. In this section, we always assume that the manifold $M$ has dimension $d\geq 2$.

Definition 8.1**.**

We call stiff pair the data of $(M,\nabla)$ where $M$ is a similarity pseudo-Euclidean manifold while $\nabla$ is a stiff connection for this structure.

An isomorphism of stiff pairs $(M,\nabla)$, $(M^{\prime},\nabla^{\prime})$ is a diffeomorphism $f:M\longrightarrow M^{\prime}$ such that:

1. (1)

in each chart, $f$ is a similarity between the two pseudo-Euclidean structures; 2. (2)

$\nabla$ is the pullback of connection $\nabla^{\prime}$ by $f$.

We also consider a variation of the notion where similarities are replaced by isometries and call them isometry stiff pairs.

We recall that a similarity is the composition of a homothety with an isometry, which must preserve the sign of $q$, but not necessarily the orientation. We also recall that in this article, we call negalitudes the affine maps $f$ such that $q(\vec{f}(v))=-r^{2}q(v)$ for some $r>0$ and all vectors $v$.

We first prove that for a given connected pair, each chart of the atlas corresponds to a unique canonical model.

Proposition 8.2**.**

For a connected stiff pair $(M,\nabla)$, one of the following statements holds:

• •

$\nabla$* is a flat connection on $M$;*

• •

there is a generating family $\mathcal{F}$ of the pseudo-Euclidean similarity atlas such that every chart of $\mathcal{F}$ is a restriction of the same canonical model (see Section 4.5).

Above, $\lambda$ is constrained to be one of $\{-1,0,1\}$. For isometry stiff pairs, the proposition still holds if we include canonical models for which $\lambda$ can be any real.

Proof.

For each chart, we know by Proposition 4.19 that for each point of $M$ exactly one of the following holds: either $\nabla$ is locally flat or there exists a local chart sending $\nabla$ to a canonical model, which is unique and nowhere flat. Using that connected sets have no non-trivial partition into open sets, the proposition follows.

In the case of isometry stiff pairs, the proof is the same, using Remark 4.20. ∎

In other words, a pair $(M,\nabla)$ with $M$ connected and $\nabla$ not flat is modeled on a single canonical model and its group of automorphisms (see the $(G,X)$-manifold setting in Section 8.1).

When $\lambda\neq 0$, the isomorphisms between small open subsets of model $\cal S_{\lambda}^{\nu}(p,d-p)$ are isometries (Lemma 4.26 and Proposition 4.27). These isometries preserve the function $q+\lambda$. Hence:

Corollary 8.3**.**

Stiff pairs that are connected, non-flat and for which $\lambda\neq 0$ are actually isometry stiff pairs. On any isometry stiff pair, a potential $\psi$ for $\nabla$ can be globally defined on $M$.

Classification of (isometry) stiff pairs is a very general problem. We will just deal with two interesting special cases:

• •

$M$ is a compact manifold;

• •

$(M,\nabla)$ is geodesically complete.

We remind the definition of being geodesically complete for arbitrary connections.

Definition 8.4**.**

A smooth manifold $M$ endowed with an affine connection $\nabla$ is geodesically complete if every geodesic of $\nabla$ is parameterized from $-\infty$ to $+\infty$.

8.1. Developing map

We start by explaining that $(M,\nabla)$ pairs naturally pull back to the universal cover $\tilde{M}$ of $M$.

Pulling back $(M,\nabla)$ pairs.

If $N$ is a topological space, $M$ a pseudo-Euclidean (similarity or not) manifold and $\phi:N\to M$ is a local homeomorphism then $N$ is naturally endowed with a pseudo-Euclidean (similarity) manifold atlas, unique up to atlas equivalence, such that $\phi$ a local isomorphism thereof: just take the composition of $\phi$ with appropriate restrictions of charts of $M$. If moreover $M$ is endowed with a stiff connection $\nabla$ then the connection $\phi^{*}\nabla$ is stiff. In particular when $M$ is connected and $N$ is a universal cover of $M$, it naturally bears the pull-back structure of $(M,\nabla)$.

$(G,X)$-manifold setting

We will use classical arguments of what W.P. Thurston calls $(G,X)$-manifolds and some other authors locally homogeneous manifolds. Our reference for the results here is [Thu97], in particular Section 3.4 in this book. Pairs $(M,\nabla)$ with $M$ connected are equivalent to the data of a $(G,X)$-manifold with $X=\cal S$ and $G$ its automorphism group. Normally, the group $G$ is required to act transitively on $X$ but in our setting this is the case if and only similarities are allowed and $\lambda=0$, so not all the classical theory of $(G,X)$-manifolds necessarily applies.

The next result does not require the action of $G$ to be transitive (according to the first paragraph of page 141 of [Thu97] after the Definition 3.4.2).

Corollary 8.5** (Developing map).**

If $(M,\nabla)$ is a stiff pair or an isometry stiff pair and $M$ connected, then there exists a map $D$ from the universal cover $\tilde{M}$ to the model $\cal S$ associated to $M$ that is a local isomorphism.

Remark 8.6*.*

This is useful for instance when considering a geodesic curve $\gamma$ of $M$, which can henceforth be entirely made to correspond to (a subset of) a geodesic of $\cal S$: more precisely for any lift $\tilde{\gamma}$ in $\tilde{M}$ of a parameterized geodesic $\gamma$ of $M$, the function $D\circ\tilde{\gamma}$ defines a parameterized geodesic of $\cal S$. One consequence is that the maximal interval of definition of $\gamma$ is contained in191919But not necessarily equal to: think of the case where $M$ is a compactly contained open subset of $\cal S$. the maximal interval of definition of $D\circ\tilde{\gamma}$.

Definition 8.7**.**

A complete $(G,X)$-manifold is one such that $D:\tilde{M}\to X$ is a covering.

Note that if $X$ is simply connected, $D$ is then actually an isomorphism. This is not to be mistaken with geodesically complete (Definition 8.4).

8.2. Deducing the pseudo-Euclidean structure from the connection

In the non-flat case, it is actually possible to deduce the pseudo-Euclidean structure from the connection.

Proposition 8.8**.**

Consider a smooth manifold $M$ endowed with a connection $\nabla$. Assume that $M$ is connected and that near every point there is a chart taking values in a pseudo-Euclidean space, whose signature is a priori allowed to vary, in which $\nabla$ is stiff. Then either:

• •

$\nabla$* is a flat connection on $M$;*

• •

there is a single canonical model $\cal S$ and a generating family $\mathcal{F}$ of the smooth atlas such that every change of charts are automorphisms of $\cal S$ (in particular this atlas turns $M$ into a pseudo-Euclidean similarity manifold) and such that on every chart of $\mathcal{F}$, $\nabla$ is mapped to the connection of $\cal S$.

In the second case, the signature and model $\cal S$ are unique up to swapping all signs, i.e.: $\cal S_{\lambda}^{\nu}(p,d-p)\leftrightarrow\cal S_{-\lambda}^{-\nu}(d-p,d)$.

Proof.

Consider the system of charts provided as a hypothesis. We can assume that all charts take values in the finite number of spaces $E^{p,d-p}$ where $p$ varies from [math] to $d$. Let $O$ be the set of points in $M$ that have a neighbourhood on which $\nabla$ is flat. For any point not in $O$, by Theorem 4.4 and Remark 4.6, a whole neighbourhood is disjoint from $O$. By connectedness of $M$, either $O=M$ or $O=\emptyset$. In first case, $\nabla$ is flat on all $M$.

We assume now we are in the second case: $O=\emptyset$. By Proposition 4.19, possibly restricting and applying a similarity, we can arrange so that all charts take value in canonical models (they are allowed to differ on different charts) and send $\nabla$ to the respective connection of the models. By Lemma 4.22 the changes of charts are either locally restrictions of similarities fixing [math] between spaces with identical signature and $\nu$ is the same, or of negalitudes between some $E^{(p,d-p)}$ and $E^{(d-p,p)}$ and then the two charts have different values of $\nu$. We can replace all charts for which $\nu<0$ by a chart for which $\nu>0$, using the map $(x_{1},\ldots,x_{d})\mapsto(x_{d},\ldots,x_{1})$. Then the new chart system $\cal F$ has only similarities as transition maps by Lemma 4.22 and the signature $(p,d-p)$ only depends on $x$ and is locally constant. By connectedness it is the same for all charts.

Lemma 4.22 is a bit more precise: for similarities it also states that the sign of $\lambda$ is the same. Since $\lambda\in\{-1,0,1\}$ in canonical models, $\lambda$ is actually the same for all charts. We thus have an atlas $\cal F$ satisfying the required properties. ∎

The ambiguity for the model raises interesting questions when the signature is $(p,p)$ and $\lambda=0$. Given a pseudo-Riemannian manifold $M$, at every point $x\in M$, the tangent space $T_{x}M$ contains a light cone $g(v,v)=0$ and the remaining vectors in $v\in T_{x}M$ split into two sets according to the sign of $g(v,v)$, which we will call the positive and the negative cones. In signature $(p,p)$, the fact that $M$ is a pseudo-Riemannian manifold means that the two cones are clearly distinguished by their sign, though they are linearly indistinguishable: they can be swapped by a negalitude. For a general smooth manifold $M$ on which a smooth field of $(p,p)$ light cones is defined, but without reference of a $g$ allowing to distinguish the positive and negative cones, one can construct examples where following a specific not null homotopic loop in $M$ identifies one cone in $T_{x}M$ with its complement. However, this cannot be done in the situation of Proposition 8.8: even though one can choose between two opposite models $\cal S$ in which $M$ sits, once one is chosen, the positive and negative cones are fixed on all tangent spaces. One may wonder if there is a way to distinguish these cones a priori, i.e. without reference to a model $\cal S$. The answer is yes: in the situation of Proposition 8.8, for any nondegenerate $2$-dimensional vector subspace $\vec{P}$ of $T_{x}M$ on which $g$ is nondegenerate, the infinitesimal holonomy for $\nabla$ of an infinitesimal loop about $x$ and contained in $\vec{P}$ will push vectors in the sense of the loop if the vector belongs to one cone, and in the opposite sense to the other. See the analysis presented at the end of Section 4.6.

8.3. Stiff connections on compact manifolds

Theorem 8.9**.**

We consider a stiff pair $(M,\nabla)$ that is connected, non-flat and compact. Then $(M,\nabla)$ cannot be an isometry stiff pair. Let $\mathcal{S}_{\lambda}^{\pm}(p,d-p)$ the associated canonical model (see Proposition 8.2). Then the pair $(M,\nabla)$ satisfies the following properties:

1. (1)

$\lambda=0$; 2. (2)

$M$* has a non-vanishing global vector field (in particular $\chi(M)=0$);* 3. (3)

$(M,\nabla)$* is not geodesically complete;* 4. (4)

there is no globally defined volume form preserved by $\nabla$.

Proof.

That $(M,\nabla)$ is an isometry stiff pair means that there is an atlas whose change of charts are isometries centered on [math]. In particular, $\psi:=q+\lambda$ is preserved by these isometries and there is a well-defined potential function $\psi$ defined on $M$. Since $M$ is compact, $\psi$ has both a maximal and a minimal point on $M$. So in the canonical model $\cal S$, $\psi$ has both a local maximum and local minimum. However, none of the canonical models have this property: indeed, only $0\in E^{p,d-p}$ can be a local extremum of $\psi$ (and only for some of the models). This leads to a contradiction.

So $(M,\nabla)$ is not an isometry stiff pair. By Corollary 8.3, $\lambda\neq 0$. (Instead of Corollary 8.3, one can use the characteristic frequency $\omega=\sqrt{\epsilon\frac{q+\lambda}{q-\lambda}}=\sqrt{\epsilon\left(1-2\frac{\lambda}{\psi}\right)}$ associated to each equipotential leaf: it can be checked that $\omega$ depends on $x\in M$ but does not depend on the chart in which it is defined, and that if $\lambda\neq 0$ then one deduces $\psi$ from $\omega$. Then the analysis of the first paragraph holds.)

Call $U$ the domain of the canonical model. Since $\lambda=0$, $U$ is disjoint from the light cone. Choose a radial vector at any point $x_{*}\in U$ and transport it by the group $G$ of similarities that preserves the domain $U$. Since $\lambda=0$, the group $G$ acts transitively on $U$. Since moreover the fixator of $x_{*}$ leaves the vector unchanged, this defines a unique vector at each point. We obtain a smooth vector field that is invariant by $G$. It thus passes to the quotient and defines a non-vanishing vector field on $M$.

Let $U$ denote the domain of canonical model $\cal S$. Since $\lambda=0$, all rays (which we recall are straight half line starting from [math]) through a point $x\in U$ are entirely contained in $U$. By section 7.4 all these rays reach infinity in finite time ($\lambda=0$ hence $q(x)\neq 0$ so $\psi=q$ has degree $2$ on the ray through [math] and only vanishes at the origin). Consequently, the canonical model is not geodesically complete. We can transfer this to $M$ as follows: consider a developing map $D$ as provided by Corollary 8.5. Consider any point $x$ of $M$ and a lift $\tilde{x}\in\tilde{M}$. The direction of the ray through $D(\tilde{x})$ towards infinity corresponds to a direction in $T_{x}M$. Consider a geodesic of $M$ starting in this direction. We only look at positive times. By the discussion in Remark 8.6, this geodesic has finite positive time of existence. So $M$ is not geodesically complete.

In the model, $\nabla$-invariant volume forms (local or global) have been identified in Proposition 3.8 as $\omega=(\alpha/\psi^{d+1})dx_{1}\wedge\cdots\wedge dx_{d}$ for $\alpha\in{\mathbb{R}}^{*}$. Since $\lambda=0$ we have $\omega=(\alpha/q^{d+1})dx_{1}\wedge\cdots\wedge dx_{d}$. Such a volume form is invariant by $\operatorname{SO}(U)$, but not invariant by homotheties $f:x\mapsto sx$ with $|s|\notin\{-1,0,1\}$: indeed

[TABLE]

Let us now assume that $M$ has a $\nabla$-invariant volume form. For any point in $M$, consider a chart taking values in $U$, and let $\alpha\in{\mathbb{R}}^{*}$ be such that this form of $M$ takes expression $\omega=(\alpha/\psi^{d+1})dx_{1}\wedge\cdots\wedge dx_{d}$ near the point. We can choose $s>0$ such that when we divide the chart by $s$, the form takes the expression $\pm 1/\psi^{d+1}dx_{1}\wedge\cdots\wedge dx_{d}$ in the new chart. We thus get a new atlas for which the invariant form has expression $\pm 1/\psi^{d+1}dx_{1}\wedge\cdots\wedge dx_{d}$ everywhere. In this new atlas, by Equation 8.1, the changes of charts are necessarily isometries. This would turn $(M,\nabla)$ into an isometry stiff pair, leading to a contradiction. ∎

In particular, for $d=2$, $M$ is either a torus or a Klein bottle (the only compact surfaces with a non-vanishing vector field). Both these two cases are easily seen to occur in any signature (recall that $\lambda=0$).

Remark 8.10*.*

Consider a canonical model $\cal S$ for which $\lambda=0$ and denote $G={{\mathbb{R}}_{+}^{*}\!\operatorname{O}}(U)$ its stiff manifold automorphism group, which acts transitively on the domain $U$ of $\cal S$. Determining whether all compact $(G,\cal S)$-manifolds are complete $(G,\cal S)$-manifolds202020The notion of complete $(G,X)$-manifolds is recalled in Section 8.1. It is not to be mistaken with geodesically complete. is more or less equivalent to determining if all compact manifolds modeled on the leaf $q=1$ or $q=-1$ with automorphism group $\operatorname{O}(U)$ are complete. Unless $p$ or $d-p$ is small, this is still an open problem (and a particular case of the Markus conjecture).

Remark 8.11*.*

Conversely, the general question of the existence of compact stiff pairs when $\lambda=0$ is probably equivalent to the existence of compact manifold quotients of the universal cover $\tilde{\cal Q}$ of the quadric $\cal Q$ of equation $q(x)=\nu$ (with $\nu=\pm 1$), considered as a $(G,\cal S)$-manifold, under its automorphism group. Note that $\tilde{\cal Q}=\cal Q$ unless ($\nu=1$ and $p=2$) or ($\nu=-1$ and $d-p=2$). This question is not yet solved for all values of $(p,d-p)$. We give below a short and incomplete list of known results—assuming $\nu=1$—(the results for $\nu=-1$ can be deduced by permuting $p$ and $d-p$ in the signature). The Calabi-Markus phenomenon (see [Wol62]) implies that if $p>d-p$ then any manifold quotient of $\cal Q$ is a finite quotient, which prevents the existence of compact quotients if $2\neq p>d-p>0$. On a different line of argument, for $p$ even, $d-p$ odd then [Kul81] proves that the Euler characteristic and the Gauss-Bonnet formula prevent the existence of compact quotients. On the other hand, for signature $(d,0)$ and we are on a Euclidean sphere, which is already compact. For signature $(1,d-1)$ the situation is equivalent to hyperbolic geometry and there are compact quotients. Similarly for signature $(2,2k)$ complex hyperbolic geometry allows to create compact quotients, and the same can be done for signatures $(4,4k)$ using quaternions.

8.4. Geodesically complete stiff connections

Models $\mathcal{S}_{-1}^{+}(d,0)$ for $d\geq 2$ are the only geodesically complete examples, as the following theorem shows.

Theorem 8.12**.**

We consider a pair $(M,\nabla)$ formed by a similarity pseudo-Euclidean manifold $M$ of dimension $d\geq 2$ and a stiff connection $\nabla$ that is non-flat. Then $(M,\nabla)$ is geodesically complete if and only if: the signature is pure, i.e. $(d,0)$ or $(0,d)$, and $(M,\nabla)$ is isomorphic to respectively the canonical model $\mathcal{S}_{-1}^{-}(d,0)$ or $\mathcal{S}_{1}^{+}(0,d)$ (i.e. the ones that are defined on bounded subsets of ${\mathbb{R}}^{d}$).

Proof.

First we show that the models above are geodesically complete. By sign reversal, it is enough to treat the case of signature $(d,0)$. The analysis done in Section 7.4 shows that $\mathcal{S}_{-1}^{-}(d,0)$ is geodesically complete: indeed every straight line in ${\mathbb{R}}^{d}$ that intersects the domain at some point $x$ of the model also intersects the unit sphere at two points, one on each side of $x$. We are thus in the first sub-case of Case 3 in the analysis performed in Section 7.4.

Conversely, we assume that $(M,\nabla)$ is geodesically complete. Then its universal cover is also geodesically complete. Consider the map $D:\tilde{M}\to\cal S$ provided by Lemma 8.5. As in the proof of Theorem 8.9, we can use Remark 8.6 and deduce that from any point in $D(M)$ and any initial direction, the geodesic is defined for an infinite amount of time.

By way of contradiction, assume that the domain of $\cal S$ is not bounded. Then for all $x$ in it, let us prove that there is at least one direction $v$ for which $q(v)\neq 0$ and for which the infinite affine half line $[x,v)$ is contained in the model, i.e. $q(x+sv)+\lambda\neq 0$ when $s>0$. Indeed $P(s):=q(x+sv)+\lambda=q(x)+\lambda+2(x\cdot v)s+q(v)s^{2}$. Choose $v$ such that $q(v)$ has the same sign as $q(x)+\lambda$: if $q$ has both positive and negative signature terms, this is obviously possible. Otherwise by Lemma 4.14 and the hypothesis that $\cal S$ is unbounded, we have either $\nu=+$ and $\sigma=(0,d)$ or $\nu=-$ and $\sigma=(d,0)$, so any $v$ fits. Then polynomial $P$ has either no real roots, or its two real roots (possibly equal) have positive product, so lie on the same side of [math]. Possibly replacing $v$ by $-v$, we satisfy the requirement that $P$ does not vanish on $[0,+\infty)$.

For the geodesic $\gamma$ such that $\gamma(0)=x$ and $\dot{\gamma}(0)=v$ with $v$ as above, the existence time would be bounded in the positive direction. It follows that the corresponding geodesic in $M$ would be only defined for bounded time in the positive direction: $M$ would not be geodesically complete, leading to a contradiction. Hence, actually, the domain of $\cal S$ is bounded.

Up to sign reversal, it is enough to treat the case when the canonical model is $\cal S=\cal S_{-1}^{-}(d,0)$. Consider again the developing map $D:\tilde{M}\to\cal S$.

Let $\tilde{x}_{0}\in M$ and $y_{0}=D(\tilde{x}_{0})$. Since $(M,\nabla)$ is geodesically complete, the geodesic exponential map $\exp_{\tilde{x}_{0}}$ is defined on the whole tangent plane $T_{\tilde{x}_{0}}\tilde{M}$. This is also true for the exponential map $\exp_{y_{0}}$ at $y_{0}$ in $\mathcal{S}$. The two exponential maps are related by the following commutative diagram, where $L$ is the differential of $D$ at $\tilde{x}_{0}$:

[TABLE]

Now note that $\exp_{y_{0}}$ is a bijection from $T_{y_{0}}\cal S$ to $\cal S$, because the model is $\cal S=\cal S_{-1}^{-}(d,0)$: it sends each vector line $\vec{L}\subset T_{y_{0}}$ bijectively on the segment that is the intersection of $y_{0}+\vec{L}$ with the unit Euclidean ball of ${\mathbb{R}}^{d}$.

It follows that $D$ has a section $\Phi=\exp_{\tilde{x}_{0}}\circ\leavevmode\nobreak\ L^{-1}\leavevmode\nobreak\ \circ\leavevmode\nobreak\ \exp_{y_{0}}^{-1}:\cal S\to\tilde{M}$. Since $D$ is a local isomorphism and satisfies $D\circ\Phi=\operatorname{Id}_{\cal S}$, section $\Phi$ is an isomorphism212121In the category of manifolds locally modeled on $\cal S$ and its automorphisms (see Section 8.1). to its image.

Since moreover $\cal S$ is geodesically complete, the image of $\Phi$ is a geodesically complete subset of $\tilde{M}$. Any two points of $\tilde{M}$ can be joined by a finite sequence of geodesics of $\tilde{M}$ and it follows that the image of $\Phi$ is all $\tilde{M}$.

We have thus proved that $\tilde{M}$ is isomorphic to $\cal S=\mathcal{S}_{-1}^{-}(d,0)$. In particular, $M$ is a quotient of $\cal S$ by a subgroup of its automorphism group that is free of fixed points and properly discontinuous. Since every automorphism of $\cal S$ preserves the origin, there is no nontrivial quotient of $\cal S$. The pair $(M,\nabla)$ is thus isomorphic to $\cal S$. ∎

We can also give an interesting characterization of canonical model $\mathcal{S}_{-1}^{-}(d,0)$.

Corollary 8.13**.**

In the $d$-dimensional open unit ball $\mathcal{B}^{d}$ in classical Euclidean space ${\mathbb{R}}^{d}$, there is a unique symmetric affine connection $\nabla$ satisfying the following conditions:

1. (1)

$(\mathcal{B}^{d},\nabla)$* is geodesically complete;*

and $\nabla$ is stiff, i.e.

1. (2)

unparameterized geodesics of $\nabla$ coincide with the usual lines of $\mathcal{B}^{d}$; 2. (3)

at each point the first order infinitesimal holonomy of $\nabla$ is an infinitesimal rotation for the usual conformal structure of $\mathcal{B}^{d}$.

Proof.

Let us denote $\|x\|$ the Euclidean norm. The pair $(\mathcal{B}^{d},\nabla)$ is a Euclidean manifold endowed with a stiff connection. Theorem 4.4 proves that in the trivially defined chart, the potential $\psi$ defining $\nabla$ has a specific form: up to multiplication by a constant, $\psi(x)=\|x\|^{2}+\Lambda(x)$ where $\Lambda:{\mathbb{R}}^{d}\to{\mathbb{R}}$ is affine. The potential should vanish on the boundary of the ball, otherwise $(\mathcal{B}^{d},\nabla)$ would not be geodesically complete. Consequently, the potential can only be $\psi(x)=\|x\|^{2}-1$ up to multiplication by a constant, and we obtain canonical model $\mathcal{S}_{-1}^{-}(d,0)$. ∎

Remark 8.14*.*

In dimension $d\geq 3$, Proposition 4.10 implies that we can weaken the hypothesis on first order infinitesimal holonomy and only require that at each point it is an infinitesimal similarity. (This was called weakly stiff in Section 4.)

Open Question 8.15**.**

In dimension $d=2$, does Corollary 8.13 hold under the weaker hypothesis that the first order infinitesimal holonomy consists in similarities?

Using the description of weakly stiff connections in dimension $2$ given in Section 4.4, we reformulate the question above at the end of the next section into an analytic question about meromorphic functions on the unit disk.

8.5. Geodesically complete stiff and weakly stiff connections on the disk

According to Corollary 8.13, the connection $\nabla$ of model $\mathcal{S}_{-1}^{-}(2,0)$ is the only geodesically complete stiff connection on the unit disk of $E^{2,0}$. In Table 1, we compare $\mathcal{S}_{-1}^{-}(2,0)$ with the usual geometric models of the hyperbolic plane on the disk:

• •

The Cayley-Klein model (where geodesics are chords);

• •

The Beltrami-Poincaré model (where the Levi-Civita connection preserves the conformal structure of $E^{2,0}$).

Recall that the hyperbolic plane is geodesically complete. The geodesics of the Cayley-Klein model are the usual straight lines and thus coincide with (unparameterized) geodesics of the canonical model $\mathcal{S}_{-1}^{-}(2,0)$. However, the isochrone metric of $\mathcal{S}_{-1}^{-}(2,0)$ does not coincide with the Cayley-Klein hyperbolic metric nor the Poincaré metric.

In contrast with what happens in dimension $d\geq 3$, weakly stiff connections in dimension two are not automatically incompressible. In Proposition 8.16, we refine Corollary 8.13 by weakening the hypothesis on infinitesimal holonomy.

Proposition 8.16**.**

Let $\nabla$ be a symmetric affine connection on the open unit disk $\mathbb{D}\subset{\mathbb{C}}$ such that $(\mathbb{D},\nabla)$ is geodesically complete and $\nabla$ is weakly stiff, i.e.

1. (1)

unparameterized geodesics of $\nabla$ coincide with the usual lines of $\mathbb{D}$; 2. (2)

at each point the infinitesimal holonomy of $\nabla$ is an infinitesimal similarity for the usual conformal structure of $\mathbb{D}$.

Denote $a_{1},a_{2}$ the coefficients of the associated form of $\nabla$ (see Corollary 3.2). Then there is a meromorphic function $f$ on $\mathbb{D}$ such that $f(z)+\bar{z}$ does not vanish and the complex-valued function $A=a_{1}+ia_{2}$ is of the form $A(z)=\frac{-2}{\bar{z}+f(z)}$. Besides, one of the two following statements holds:

1. (A)

$f(z)=-\frac{1}{z}$, i.e. we have model $\cal S^{-}_{-1}(2,0)$; 2. (B)

$f$* does not extend to any point222222By $f$ extends to $z\in\partial\mathbb{D}$ we mean that there exists a meromorphic map defined on $\mathbb{D}\cup B(z,\epsilon)$ whose restriction to $\mathbb{D}$ is $f$. of the boundary circle of $\mathbb{D}$.*

Proof.

The existence of $f$ follows from Proposition 4.12. If $f$ extends as a meromorphic function to a neighbourhood $V$ of some point $z_{0}$ of the boundary circle $\mathcal{C}$, $\bar{z}+f(z)$ should vanish for all $z\in V\cap\mathcal{C}$ otherwise $(\mathbb{D},\nabla)$ would fail to be geodesically complete. Thus, there is an arc of $\mathcal{C}$ where $f$ extends and coincides with $-\frac{1}{z}$. The isolated zeroes theorem then implies that $f$ globally coincides with $-\frac{1}{z}$. ∎

Can Case B of Proposition 8.16 happen? If $f$ is a meromorphic function defined on the unit disk, such that $f(z)+\bar{z}$ does not vanish, and such that $\mathbb{D}$ is its maximal domain of holomorphy, it is not obvious to figure out whether or not all geodesics of the associated $\nabla$ take an infinite amount of time to reach $\partial\mathbb{D}$. Using Section 3.6, we can reformulate this purely in terms of $f$.

Open Question 8.17**.**

Does there exist a meromorphic function $f$ on $\mathbb{D}$ such that:

• •

$\mathbb{D}$ is the maximal domain of holomorphy of $f$,

• •

$f(z)+\bar{z}$ does not vanish on $\mathbb{D}$,

• •

on any straight chord of $\mathbb{D}$, the following integral along the chord diverges on both sides of the chord:

[TABLE]

where $F$ is defined as the following antiderivative along the chord:

[TABLE]

The answer to Question 8.15 is positive if and only if the answer to Question 8.17 is negative.