A general version of Beltrami's theorem in Finslerian setting

TL;DR

This paper provides a new proof of a Finslerian version of Beltrami's theorem, extending its applicability to all dimensions including dimension 2, thereby broadening the understanding of Finsler geometry.

Contribution

It introduces a novel proof of the Finslerian Beltrami's theorem that is valid in any dimension, including the two-dimensional case.

Findings

New proof of Finslerian Beltrami's theorem applicable in all dimensions

Extension of classical theorem to Finsler geometry in dimension 2

Broader understanding of geometric properties in Finsler spaces

Abstract

We present a new proof of a Finslerian version of Beltrami's theorem (1865) which works also in dimension 2.

Equations

$$h=\frac{1}{2}(\mathrm{Id}-[S,J]),v=\frac{1}{2}(\mathrm{Id}+[S,J]).$$

$$\Phi :=v\circ [S,h],R:=\frac{1}{2}[h,h],$$

$$3R=[J,\Phi ],\Phi =i_{S}R.$$

$$\rho :=\frac{1}{n-1}\mathrm{Tr}(\Phi ),$$

$$\Phi =\rho J-\alpha \otimes \mathcal{C}.$$

$$R=\xi \wedge J-d_J\xi \otimes \mathcal{C},$$

$$3R=[J,\Phi ]=[J,\rho J-\alpha \otimes \mathcal{C}]=(d_J\rho +\alpha )\wedge J-d_J\alpha \otimes \mathcal{C},$$

$$\xi :=\frac{1}{3}\left(\alpha +d_J\rho \right).$$

$$\Phi =i_{S}R=i_S\left(\xi \wedge J-d_J\xi \otimes \mathcal{C}\right)=i_S\xi J-(i_S{d}_J\xi +\xi )\otimes \mathcal{C}.$$

$$0=[h,R]=[h,\xi \wedge J-d_J\xi \otimes \mathcal{C}]=d_h\xi \wedge J-d_h{d}_J\xi \otimes \mathcal{C}.$$

$$(n-2)d_h\xi -i_S{d}_h{d}_J\xi =0.$$

$$-i_S{d}_h{d}_J\xi =i_S{d}_J{d}_h\xi =-d_J{i}_S{d}_h\xi +{\mathcal{L}}_{JS}{d}_h\xi +i_{[J,S]}{d}_h\xi ={\mathcal{L}}_{\mathcal{C}}{d}_h\xi +i_{\mathrm{Id}-2v}{d}_h\xi =d_h\xi +2d_h\xi =3d_h\xi .$$

$$\Phi =\kappa (F^{2}J-F{d}_{J}F\otimes \mathcal{C}).$$

$$\rho =\kappa F^2\text{and}\alpha =\kappa F{d}_{J}F,$$

$$\xi =\frac{1}{3F}{d}_J\left(\kappa F^3\right)=\frac{F^2}{3}{d}_J\kappa +\kappa F{d}_{J}F.$$

$$R=\kappa F{d}_{J}F\wedge J.$$

$$d_J\xi =0;$$

$$d_h\xi =0$$

$$0=d_J\xi =\frac{2F}{3}{d}_{J}F\wedge d_J\kappa +F{d}_J\kappa \wedge d_{J}F=\frac{F}{3}{d}_J\kappa \wedge d_{J}F,$$

$$d_J\kappa \wedge d_{J}F=0.$$

$$0=d_h\xi =d_h(\kappa F{d}_{J}F)=F{d}_h\kappa \wedge d_{J}F+\kappa F{d}_h{d}_{J}F=F{d}_h\kappa \wedge d_{J}F,$$

$$d_h\kappa \wedge d_{J}F=0.$$

$$S(\kappa )d_{J}F-F{d}_h\kappa =0.$$

$$d_J{\mathcal{L}}_S\kappa ={\mathcal{L}}_S{d}_J\kappa +d_{[J,S]}\kappa =d_{h-v}\kappa =d_h\kappa ,$$

$$S(\kappa )d_{J}F-F{d}_{J}S(\kappa )=0\iff S(\kappa )\frac{d_{J}F}{F^2}-\frac{1}{F}{d}_{J}S(\kappa )=0\iff d_J\left(\frac{S(\kappa )}{F}\right)=0.$$

$$d_h{d}_{J}P=0.$$

$$\tilde{\xi }=\xi +P{d}_{J}P-d_{h}P.$$

$$d_J\tilde{\xi }=d_J\xi -d_J{d}_{h}P.$$

$$d_{\tilde{h}}\tilde{\xi }$$

$$=d_{h-PJ-d_{J}P\otimes \mathcal{C}}\xi +d_{h-PJ-d_{J}P\otimes \mathcal{C}}P{d}_{J}P-d_{h-PJ-d_{J}P\otimes \mathcal{C}}{d}_{h}P$$

$$=d_h\xi -P{d}_J\xi -d_{J}P\wedge \xi +d_{h}P\wedge d_{J}P+P{d}_h{d}_{J}P-d_{R}P+P{d}_J{d}_{h}P+d_{J}P\wedge d_{h}P$$

$$=d_h\xi -P{d}_J\xi +\xi \wedge d_{J}P-d_{R}P+P{d}_J{d}_{h}P+P{d}_h{d}_{J}P$$

Full text

1. Introduction

Flag curvature in Finsler geometry is a natural extension of sectional curvature. While in Riemannian geometry, metrics of constant sectional curvature are well understood and classified, in Finsler geometry the problem is far from being solved, [10, 16]. In Finsler geometry, there are many characterisations for metrics of constant flag curvature, [1, 2, 3, 4]. In this paper, in Theorem 3.1, we characterise Finsler metrics of constant curvature in terms of three conditions, which we call CC-conditions: (3.5), (3.6), (3.7). There are two motivations for this formulation of the three CC-conditions. First, we can view Theorem 3.1 as a Finslerian extension of Schur’s lemma that includes dimension $2$ as well. Secondly, it is easy to check whether or not each of the three CC-conditions are projectively invariant.

The first and third CC-condition are invariant under projective deformations and they have corresponding quantities in the Riemannian context. The second CC-condition is not projectively invariant unless the projective factor is a Hamel function. Therefore, the second CC-condition gives, for a Finsler metric of constant curvature, the class of projectively equivalent Finsler metrics that also have constant curvature.

Beltrami’s theorem states that a Riemannian metric projectively equivalent to a Riemannian metric of constant curvature has also constant curvature, [14, vol. IV, p.19]. A classic proof of Beltrami’s theorem uses the projective Weyl tensor if $n>2$ and the Liouville (Cotton) tensor if $n=2$, see [13] for a recent survey on the projective geometry of affine sprays.

In this work we prove the following general version of Beltrami’s theorem, in Finslerian setting, for dimension $n>1$.

If two Finsler metrics over the same connected manifold, of dimension $n>1$, have projectively equivalent geodesic sprays and one is of constant flag curvature then the other is also of contant curvature if and only if the projective factor is a Hamel function (Theorem 3.2).

For the proof of Theorem 3.2, we study the projective invariance of the three CC-conditions proposed in Theorem 3.1; the first two conditions for $n>2$ and the last two conditions for $n=2$. The projective invariant $2$-form from the third CC-condition can be viewed as the Cotton (Liouville) tensor of the spray. The Cotton tensor has been introduced very recently by Crampin in the geometry of a spray, [7], as an obstruction for $2$-dimensional sprays to be projectively $R$-flat.

In dimension $n>2$, the Finslerian version of Beltrami’s theorem has also been proved using a Weyl-type curvature tensor that characterises Finsler metrics of constant curvature in [3, Theorem 5.2] and an algebraic characterisation for Finsler metrics of constant curvature in [4].

2. Isotropic sprays

We start this section with a brief overview of the geometric setting associated to a spray. We consider $M$ a smooth, $n$-dimensional, connected manifold, with $n>1$. $TM$ is the tangent bundle and $T_{0}M=TM-\{0\}$ is the slit tangent bundle of $M$. On $TM$ there are two canonical structures that we will use further on, the Liouville (dilation) vector field $\mathcal{C}$ and the vertical endomorphism $J$.

A spray $S$ is a special vector field on $T_{0}M$ such that $JS=\mathcal{C}$ and $[\mathcal{C},S]=S$. A geodesic of a spray is a smooth curve on $M$ whose velocity is an integral curve of the spray.

An orientation preserving reparametrisation $t\mapsto\widetilde{t}(t)$ of the geodesics of a spray $S$ leads to the geodesics of a new spray $\widetilde{S}=S-2P\mathcal{C}$, where $P\in C^{\infty}(T_{0}M)$ is positively 1-homogeneous.

Definition 2.1**.**

Two sprays $S$ and $\widetilde{S}$ are projectively related if their geodesics coincide up to an orientation preserving reparametrisation.

In this work we will use the Frölicker-Nijenhuis formalism to associate geometric data to a spray $S$ [8, 9, 17, 18]. The first of them is a canonical nonlinear connection, that determines a horizontal and a vertical projector:

[TABLE]

The Jacobi endomorphism and the curvature of (the nonlinear connection determined by) $S$ are defined by

[TABLE]

respectively. They are related by

[TABLE]

The Ricci scalar $\rho\in C^{\infty}(T_{0}M)$ is given by

[TABLE]

where “Tr” means semi-basic trace (see, e.g., [18, p.134]).

Definition 2.2**.**

A spray $S$ is said to be isotropic if there exists a semi-basic 1-form $\alpha\in\Lambda^{1}(T_{0}M)$ such that the Jacobi endomorphism can be written as follows:

[TABLE]

For an isotropic spray $S$, due to (2.2), we have that $\Phi(S)=0$ and hence $\rho=i_{S}\alpha.$

The formulae (2.2) allow to reformulate the isotropy condition (2.4) in terms of the curvature tensor $R$.

Lemma 2.3**.**

A spray is isotropic if and only if its curvature tensor is of the form

[TABLE]

for a semi-basic $1$-form $\xi$ on $T_{0}M$.

Proof.

Assume first that $S$ is isotropic and hence its Jacobi endomorphism $\Phi$ is given by formula (2.4). Then, using formulae (18) and (22) in [18], we find

[TABLE]

which gives the desired equality with the the semi-basic $1$-form

[TABLE]

Conversely, assume that the curvature tensor of $S$ can be given by formula (2.5). Then

[TABLE]

Taking semi-basic trace and using formula (26) in [18] we find that $\operatorname{Tr}(\Phi)=(n-1)i_{S}\xi$. Thus the Ricci scalar is given by $\rho=i_{S}\xi$, which concludes the proof. ∎

The above characterisation of isotropic sprays has been used by Crampin in [6, 7]. We will call the semi-basic $1$-form $\xi$ in formula (2.5) the curvature $1$-form of the isotropic spray.

Lemma 2.4** (Differential Bianchi identity for the curvature $1$-form).**

In dimension $n>2$, the curvature $1$-form of an isotropic spray satisfies $d_{h}\xi=0$.

Proof.

If $S$ is any spray over $M$, then we have $[h,{\mathcal{C}}]=0$, $[h,J]=0$ (the induced connection is homogenous and torsion-free), $[h,R]=0$ (Bianchi identity); see, e.g., [8] and [17]. Now suppose that $S$ is isotropic. Then, using the just mentioned properties of the induced connection and applying formula (22) in [18], we get

[TABLE]

Applying formulae (26) and (27) in [18], we take the semi-basic trace of the right-hand side to obtain

[TABLE]

Acting by the operator $i_{S}$ on both sides of the last equality, we find that $(n-2)i_{S}d_{h}\xi=0$, whence $i_{S}d_{h}\xi=0$. Taking this into account, we can manipulate the second expression on the left hand side of (2.8) as follows

[TABLE]

This concludes the proof. ∎

The semi-basic $2$-form $d_{h}\xi$ appears in [6, 7] as an obstruction for isotropic sprays to be projectively $R$-flat. For $n>2$ this obstruction is trivial.

3. Finsler metrics of constant flag curvature and their

projective deformation

In this section we provide three necessary and sufficient conditions (CC-conditions: (3.5), (3.6) and (3.7)) for a Finsler metric to have constant curvature (Theorem 3.1). By studying the projective invariance of these CC-conditions, we arrive at a Finslerian version of Beltrami’s theorem for dimension $n>1$ (Theorem 3.2).

A Finsler metric $F$ determines a unique spray $S$ on $T_{0}M$, called its geodesic (or canonical) spray (see, e.g., [17, p. 541]). We derive the geometric data of a Finsler metric from its geodesic spray.

A Finsler metric has scalar flag curvature if there exists a [math]-homogeneous function $\kappa\in C^{\infty}(T_{0}M)$ such that the Jacobi endomorphism can be expressed as follows:

[TABLE]

For a Finsler metric of scalar flag curvature the geodesic spray is isotropic; the Ricci scalar $\rho$ and the semi-basic $1$-form $\alpha$ are given by

[TABLE]

respectively. In view of Lemma 2.3 and formulae (3.2) and (2.7), a Finsler metric has scalar flag curvature if and only if its curvature tensor is of the form (2.5) with curvature $1$-form

[TABLE]

If the Finsler metric $F$ has constant curvature (i.e., the function $\kappa$ is constant), then $\xi=\kappa Fd_{J}F$, $d_{J}\xi=0$, so its curvature tensor reduces to

[TABLE]

Theorem 3.1** (Finslerian version of Schur’s lemma for dimension $n>1$).**

Let $S$ be the geodesic spray of a Finsler metric $F$. Then $F$ has constant curvature if and only if

[TABLE]

and the curvature 1-form satisfies

[TABLE]

Proof.

Assume first that $F$ is of constant curvature. Then, as we just have seen, the geodesic spray is isotropic and $d_{J}\xi=0$. It remains to show that (3.7) is also satisfied. Since $d_{h}F=0$ and $[h,J]=0$, we have $d_{h}\xi=\kappa Fd_{h}d_{J}F=\kappa Fd_{J}d_{h}F=0$, as wanted.

Conversely, suppose that the three CC-conditions are satisfied. Then (3.5) and Lemma 8.2.2 in [15] (or Proposition 9.4.9 in [17]) imply that $F$ has scalar flag curvature $\kappa$, and hence the curvature $1$-form is given by (3.3). To prove the constancy of $\kappa$, it is sufficient to show that $d_{J}\kappa=0$ and $d_{h}\kappa=0$.

By our condition (3.6),

[TABLE]

whence

[TABLE]

Applying $i_{S}$ to both sides of (3.8), we get $(i_{S}d_{J}\kappa)d_{J}F-(i_{S}d_{J}F)d_{J}\kappa={\mathcal{C}}(\kappa)d_{J}F-{\mathcal{C}}(F)d_{J}\kappa=-Fd_{J}\kappa$, therefore $d_{J}\kappa=0$, and hence $\xi=\kappa Fd_{J}F$. Now, by condition (3.7),

[TABLE]

whence

[TABLE]

Continuing as above, we find that

[TABLE]

Since

[TABLE]

we can rewrite (3.10) as follows:

[TABLE]

Arguing by contradiction, suppose that $S(\kappa)$ is a nonzero function. Since $d_{J}\kappa=0$, $\kappa$ is a vertical lift, i.e., $\kappa=f^{v}$, for some smooth function $f$ on $M$. Then $S(\kappa)=S(f^{v})=f^{c}$ and, by the above equality, $d_{J}(f^{c}/F)=0$. Here $f^{c}$ denotes the complete lift of $f\in C^{\infty}(M)$ (see, e.g., [17, Definition 4.1.3] for the definitions of vertical and complete lifts of functions). Thus, $f^{c}/F=h^{v}$, and $F=f^{c}/h^{v}$, for some smooth function $h$ on $M$. Since $f^{c}$ is fiberwise linear, $h^{v}$ is fiberwise constant, it follows that $F$ is a fiberwise linear function. We arrived at a contradiction, hence $S(\kappa)=0$. Going back to formula (3.10), we obtain $d_{h}\kappa=0,$ which assures that $\kappa$ is constant and therefore the Finsler metric has constant curvature.

It is well known that any two-dimensional spray manifold is isotropic (see, e.g., [15, Lemma 8.1.10]), so in this case the first CC-condition is automatically satisfied.

According to Lemma 2.4, in dimension $n>2$, the curvature $1$-form of an isotropic spray also automatically satisfies the third CC-condition. ∎

The first two CC-conditions provide an equivalent characterisation for Finsler metrics of isotropic curvature (scalar curvature does not depend on the fiber coordinates) that were studied in [11]. These conditions were used to define a Weyl-type curvature tensor in [3, (4.1)] that characterises Finsler metrics of constant curvature in dimension $n>2$.

When $n>2$, there are alternative proofs of Theorem 3.1 in Finsler geometry, [12, Proposition 26.1], [17, Theorem 9.4.11].

Next, we study the projective invariance of the three CC-conditions. The isotropy condition is known to be invariant and we prove that the third CC-condition is also invariant. As to the second CC-condition, we show that it is invariant only for those projective deformations $P$ satisfying the Hamel equation

[TABLE]

A $1^{+}$-homogeneous, smooth function $P$ on $T_{0}M$, which satisfies the equation (3.11) is called a Hamel function.

Theorem 3.2** (Finslerian version of Beltrami’s theorem for dimension $n>1$).**

If two Finsler metrics over the same connected manifold have projectively equivalent geodesic sprays and one is of constant flag curvature then the other is also of constant curvature if and only if the projective factor is a Hamel function.

Proof.

Let $S$ and $\widetilde{S}$ be the geodesic sprays of two projectively related Finsler metrics. Then $\widetilde{S}=S-2P{\mathcal{C}}$, for a $1^{+}$-homogeneous, smooth function $P$ on $T_{0}M$.

The isotropy condition is invariant under projective deformations and therefore both $S$ and $\widetilde{S}$ are isotropic. According to [5, Proposition 4.4], the corresponding curvature $1$-forms $\xi$ and $\widetilde{\xi}$ are related by

[TABLE]

If we apply $d_{J}$ to both sides of this equality, we have

[TABLE]

To prove the projective invariance of the third CC-condition, we use that the horizontal projectors are related by $\widetilde{h}=h-PJ-d_{J}P\otimes{\mathcal{C}}$, see [5, (4.8)], and hence

[TABLE]

Now, using the form (2.5) of the tensor $R$, we have

[TABLE]

which together with (3.14) allows us to conclude that

[TABLE]

According to Theorem 3.1, the Finsler metric $F$ has constant curvature if and only if the three CC-conditions are satisfied: $S$ is isotropic, $d_{J}\xi=0$ and $d_{h}\xi=0$. Similarly, the projectively related Finsler metric $\widetilde{F}$ has constant curvature if and only if it satisfies the corresponding three CC-conditions. In view of formulae (3.13) and (3.15) this is true if anf only if $d_{h}d_{J}P=0$, which means that $P$ is a Hamel function. ∎

Formula (3.15) shows that the curvature $2$-form $d_{h}\xi$ is a projective invariant of isotropic sprays. According to Lemma 2.4, this curvature $2$-form vanishes in dimension $n>2$ and therefore it is a useful projective invariant for $2$-dimensional sprays, which are always isotropic. This corresponds to the Liouville (Cotton) projective invariant for affine sprays in $2$-dimensional Riemannian geometry, [13]. In Finsler geometry, Crampin [7] defines a projective Cotton tensor for arbitrary sprays, which reduces to the curvature $2$-form $d_{h}\xi$ for isotropic sprays.

Acknowledgments

We express our thanks to Mike Crampin and József Szilasi for their comments and suggestions on this work. We owe this version of the manuscript to the carefull reading and editorial suggestions of József Szilasi.

This work has been supported by Ministry of Research and Innovation within Program 1 – Development of the national RD system, Subprogram 1.2 – Institutional Performance – RDI excellence funding projects, Contract no.34PFE/19.10.2018.