Projectively and affinely invariant PDEs on hypersurfaces

TL;DR

This paper applies a general method to construct invariant PDEs on hypersurfaces in projective and affine spaces, establishing a correspondence with invariant hypersurfaces in a space of trace-free cubic forms.

Contribution

It extends a previous method to specific homogeneous spaces, providing a classification of invariant PDEs via hypersurfaces in trace-free cubic form spaces.

Findings

Invariant PDEs correspond to hypersurfaces in trace-free cubic form space.

Provides local descriptions of the invariant PDEs.

Establishes a one-to-one correspondence between PDEs and hypersurfaces.

Abstract

In [Alekseevsky, Gutt, Manno, Moreno: "A general method to construct invariant PDEs on homogeneous manifolds", Communications in Contemporary Mathematics (2021)] the authors have developed a method for constructing $G$-invariant PDEs imposed on hypersurfaces of an $(n+1)$-dimensional homogeneous space $G/H$, under mild assumptions on the Lie groups $G$. In the present paper the method is applied to the case when $G=\mathsf{PGL}(n+1)$ or $G=\mathsf{Aff}(n+1)$ and the homogeneous space $G/H$ is the $(n+1)$-dimensional projective $\mathbb{P}^{n+1}$ or affine $\mathbb{A}^{n+1}$ space, respectively. The paper's main result is that projectively or affinely invariant PDEs with $n$ independent and one unknown variables are in one-to-one correspondence with $\mathsf{CO}(d,n-d)$-invariant hypersurfaces of the space of trace-free cubic forms in $n$ variables. Local descriptions are also provided.

Tables

Table 1. Table 1. Invariant hypersurfaces for n = 3 𝑛 3 n=3 and d ≤ 10 𝑑 10 d\leq 10

degree $d$	polynomial $f$
2	$f=f_2$
4	$f=a{f}_2^2+b{f}_4$
6	$f=a{f}_2^2+b{f}_2{f}_4+c{f}_6$
8	$f=a{f}_2^4+b{f}_2^2{f}_4+c{f}_2{f}_6+d{f}_4^2$
10	$f=a{f}_2^5+b{f}_2^3{f}_4+c{f}_2^2{f}_6+d{f}_4{f}_6+e{f}_{10}$

Equations

$$(u,\mathbf{x})=(u,x^1,\dots ,x^n)$$

$$S_{f}:=\{\big{(}f(\boldsymbol{x})\,,\boldsymbol{x}\big{)}\}=\{u=f(\boldsymbol{x})\}\,.$$

$$J^{\ell }(n,M):=\bigcup _{\mathbf{p}\in M}\{[S{]}_{\mathbf{p}}^{\ell }\mid \text{S is a hypersurface of M passing through p}\}$$

$$J^{\ell }:=J^{\ell }(n,M).$$

$${\pi }_{\ell ,m}:J^{\ell }⟶J^m,[S{]}_{\mathbf{p}}^{\ell }⟼[S{]}_{\mathbf{p}}^m,\ell >m,$$

$$\cdots ⟶J^{\ell }⟶J^{\ell -1}⟶\cdots ⟶J^1=\mathbb{P}{T}^{\ast }M⟶J^0=M.$$

$$J_{a^m}^{\ell }:={\pi }_{\ell ,m}^{-1}(a^m).$$

$$0⟶V\stackrel{T}{⟶}\mathrm{Aff}(V)\stackrel{L}{⟶}\mathrm{GL}(V)⟶0$$

$$L(g):=d_{o}g:T_{o}V⟶T_{g\cdot o}V$$

$$\mathrm{Aff}(V)=T_V⋊\mathrm{GL}(V)$$

$$0⟶T_W:=T^{-1}(H)\stackrel{T}{⟶}H\stackrel{L}{⟶}{L}_H:=L(H)⟶0$$

$$H=T_W⋊L_H$$

$$h=T_{h\cdot o}\cdot (T_{h\cdot o}^{-1}\cdot h),$$

$$h=T_h\cdot L_h.$$

$$L_{h}=\left(\begin{array}[]{cc}*&*\\ 0&\overline{L}_{h}\end{array}\right)\,.$$

$$g:J^{\ell }$$

$$o^{\ell }=[S{]}_o^{\ell }$$

$$H^{(\ell )}:=G_{o^{\ell }}.$$

$${\check{J}}^{k-1}:=G\cdot o^{k-1}=G/H^{(k-1)}\subset J^{k-1}$$

$$W^k:=\tau (H^{(k-1)})\cdot o^k\subset J_{o^{k-1}}^k$$

$$\tau :H^{(k-1)}\to \mathrm{Aff}(J_{o^{k-1}}^k)$$

$$\tau (H^{(k-1)})=T_{W^k}⋊L_{H^{(k-1)}},$$

$$(u,\mathbf{x},\dots ,u_i,\dots ,u_{ij},\dots ,u_{i_1\cdots i_l},\dots )=(u,x^1,\dots ,x^n,\dots ,u_i,\dots ,u_{ij},\dots ,u_{i_1\cdots i_l},\dots )$$

$$u_{i_1\cdots i_k}\left([S_f{]}_{\mathbf{p}}^{\ell }\right)={\partial }_{i_1\cdots i_k}^{k}f(\mathbf{x}),\mathbf{p}=(u,\mathbf{x}),k\le \ell .$$

$$S^{(\ell )}:=\{[S{]}_{\mathbf{p}}^{\ell }|\mathbf{p}\in S\}.$$

$$\left(u=f(\mathbf{x}),\mathbf{x},\dots u_i=\frac{\partial f}{\partial x^i}(\mathbf{x}),\dots u_{ij}=\frac{{\partial }^{2}f}{\partial x^i\partial x^j}(\mathbf{x}),\dots \right).$$

$$T_{a^{\ell -1}}{S}^{(\ell -1)}\subset T_{a^{\ell -1}}{J}^{\ell -1},a^{\ell -1}:={\pi }_{\ell ,\ell -1}(a^{\ell }).$$

$${\mathcal{T}}^{\ell }=\left\{(a^{\ell },v)\in J^{\ell }\times T{J}^{\ell -1}|v\in T_{a^{\ell -1}}{S}^{(\ell -1)}\right\}.$$

$$D_i^{(\ell )}:={\partial }_{x^i}+\sum _{k=1}^{\ell }\sum _{j_1\le \cdots \le j_{k-1}}{u}_{j_1\dots j_{k-1}i}{\partial }_{u_{j_1\dots j_{k-1}}},i=1\dots n,$$

$${\mathcal{T}}_{[S{]}_{\mathbf{p}}^1}:={\mathcal{T}}_{[S{]}_{\mathbf{p}}^1}^1=T_{\mathbf{p}}S.$$

$$\mathcal{N}_{[S]_{\boldsymbol{p}}^{1}}:=N_{\boldsymbol{p}}S=T_{\boldsymbol{p}}M\big{/}T_{\boldsymbol{p}}S\,$$

Full text

MSC 2020: 53C30, 58J70, 35A30, 58A20

1. 1 A general construction of $G$–invariant PDEs on a homogeneous manifold $M=G/H$
2. 2 Stabilizers of the $\operatorname{\mathsf{SL}}(n+2)$–action on $J^{\ell}(n,\mathbb{P}^{n+1})$
3. 3 Stabilizers of the $\operatorname{\mathsf{Aff}}(n+1)$–action on $J^{\ell}(n,\mathbb{A}^{n+1})$
4. 4 $\operatorname{\mathsf{PGL}}(n)$– and $\operatorname{\mathsf{Aff}}(n)$–invariant PDEs on hypersurfaces of $\mathbb{P}^{n+1}$ and $\mathbb{A}^{n+1}$

Introduction

In this paper we go on constructing $G$–invariant PDEs in one unknown variable defined on $(n+1)$–dimensional $G$–homogeneous manifolds, following the general theoretical scheme developed by the authors in [1]: there the cases when $G$ is either the Euclidean $\operatorname{\mathsf{SE}}(n+1)$, or the conformal group $\operatorname{\mathsf{CO}}(n+1)$, were treated: here, we will deal with the cases when $G$ is either the projective $\operatorname{\mathsf{PGL}}(n+1)$, or the affine group $\operatorname{\mathsf{Aff}}(n+1)$.

The reason why we treat these last two cases together in a separate paper is that, unlike the two before, they give rise to third–order invariant PDEs; in particular, this casts an important bridge with the differential geometry of affine hypersurfaces and, in particular, with the Fubini–Pick invariant. On this concern, see, e.g., [2, Section 2.2], [4, Section 1], [5, Section 3.5], [9], as well as the original work of Blaschke [3].

The vanishing of this invariant defines a $G$–invariant $3^{\textrm{rd}}$ order PDE that can be constructed, according to the general scheme developed in [1], by a suitable choice of a fiducial hypersurface of order 3. In view of the tight relationship between the affine and the projective case (see also [8]), we will state a result concerning both in Section 4; technical computations concerning the projective case, that is when $\mathbb{P}^{n+1}$ is regarded as homogeneous space of $\operatorname{\mathsf{SL}}(n+2)$, will be carried out in Section 2. Analogous computations for the affine case, that is when $\mathbb{A}^{n+1}$ is regarded as homogeneous space of the affine group $\operatorname{\mathsf{Aff}}(n+1)$, which can be thought of as a “restriction” of the projective case, will be carried out in Section 3.

The main result is Theorem 4.1: each projectively or affinely invariant PDE imposed on hypersurfaces of the $(n+1)$–dimensional projective or affine space is uniquely given by a $\operatorname{\mathsf{CO}}(d,n-d)$–invariant hypersurface of the space of trace–free cubic forms in $n$ variables, were symbol $\operatorname{\mathsf{CO}}(d,n-d)$ denotes the Lie group of conformal transformations of the space $\mathbb{R}^{n}$, equipped with a metric of signature $(d,n-d)$.

A local coordinate description for the $\operatorname{\mathsf{Aff}}(n+1)$–case, which obviously works for the $\operatorname{\mathsf{PGL}}(n+1)$–case as well, will be given in Section 4.2, whereas in the last Section 4.4 we focus on the case $n=2$.

Notations and conventions

The symmetric product will be denoted by $\odot$, and the symmetric $\ell$–power of a vector space $V$ will be denoted by $S^{\ell}(V)$. If $f:M\to N$ is a differentiable map, then pull–back via $f$ of a bundle $\pi:E\to N$, is denoted by $f^{*}(E)$.

Acknowledgments

G. Manno gratefully acknowledges support by the project “Connessioni proiettive, equazioni di Monge-Ampère e sistemi integrabili” (INdAM), “MIUR grant Dipartimenti di Eccellenza 2018-2022 (E11G18000350001)”, “Finanziamento alla Ricerca” 53_RBA17MANGIO, and PRIN project 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” (code 2017JZ2SW5). G. Manno is a member ofGNSAGA of INdAM. G. Moreno is supported by the Polish National Science Centre grant under the contract number 2016/22/M/ST1/00542, as well as by the National Science Center project “Complex contact manifolds and geometry of secants”, 2017/26/E/ST1/00231.

1. A general construction of $G$–invariant PDEs on a homogeneous manifold $M=G/H$

We will review here, without proofs, the main definitions and results, as well as all the necessary preliminary material, contained in [1, Sections 2 and 3]. Throughout this section $M=G/H$ will be an $(n+1)$–dimensional homogeneous manifold and $S\subset M$ an embedded hypersurface of $M$; in Section 2 and Section 3, $M$ will be either the projective space $\mathbb{P}^{n+1}$, or the affine space $\mathbb{A}^{n+1}$, respectively.

1.1. Preliminary definitions

Locally, in an appropriate local chart

[TABLE]

of $M$, the hypersurface $S$ can be described by an equation $u=f(\boldsymbol{x})=f(x^{1},\dots,x^{n})$, where $f$ is a smooth function of the variables $x^{1},\dots,x^{n}$, that we refer to as the independent variables, to distinguish them from the remaining coordinate $u$, that is the dependent one.111A reader who is familiar with the standard literature about jet spaces may have noticed that we reversed the order of $\boldsymbol{x}$ and $u$: this choice will be more convenient for us as the coordinate $u$ will play the role of the “$0^{\textrm{th}}$ coordinate”.

We say that such a chart is admissible for $S$ or, equivalently, that the hypersurface $S$ is (locally) admissible for the chart $(u,\boldsymbol{x})$. We denote by $S_{f}=S$ the graph of $f$:

[TABLE]

Given two hypersurfaces $S_{1}$ and $S_{2}$ through a common point $\boldsymbol{p}$, one can always choose a chart $(u,\boldsymbol{x})$ about $\boldsymbol{p}$ that is admissible for both: $S_{1}=S_{f_{1}}$, $S_{2}=S_{f_{2}}$.

Definition 1.1**.**

Two hypersurfaces $S_{f_{1}},\,S_{f_{2}}$ passing through a common point $\boldsymbol{p}=(u,\boldsymbol{x})$ are called $\ell$–equivalent at $\boldsymbol{p}$ if the Taylor expansions of $f_{1}$ and $f_{2}$, in a chart admissible for both, coincide at $\boldsymbol{x}$ up to order $\ell$. The class of $\ell$–equivalent hypersurfaces to a given hypersurface $S$ at the point $\boldsymbol{p}$ is denoted by $[S]^{\ell}_{\boldsymbol{p}}$, and the union

[TABLE]

of all these equivalence classes is the space of $\ell$–jets of hypersurfaces of $M$.

Note that $J^{1}(n,M)=\mathbb{P}T^{*}M$, that is the Grassmanian bundle $\operatorname{Gr}_{n}(TM)$ of tangent $n$–planes to the $(n+1)$–dimensional manifold $M$. From now on, when there is no risk of confusion, we let

[TABLE]

The natural projections

[TABLE]

define a tower of bundles

[TABLE]

It is well known that $\pi_{\ell,\ell-1}$ are affine bundle for $\ell\geq 2$. For any $a^{m}\in J^{m}$, the fiber of $\pi_{\ell,m}$ over $a^{m}$ will be denoted by the symbol

[TABLE]

Definition 1.2**.**

A system of $m$ PDEs of order $k$ is an $m$–codimensional submanifold $\mathcal{E}\subset J^{k}$. A solution of the system $\mathcal{E}$ is a hypersurface $S\subset M$ such that $S^{(k)}\subset\mathcal{E}$.

1.2. Assumptions on the Lie group $G$

Before introducing the conditions the Lie group $G$ will have to fulfill (see Section 1.2.2 below) in order to make Theorem 1.1 work, we recall some basic facts about the affine group that will help understand the meaning of these conditions.

1.2.1. Affine groups and their subgroups of affine type

Let $V$ be a vector space, treated as an affine space: then the group $\operatorname{\mathsf{Aff}}(V)$ of affine transformations of $V$ fits into the short exact sequence of groups:

[TABLE]

The monomorphism $T$ maps a vector $v\in V$ into the corresponding parallel translation $T_{v}$: one has then a canonical normal subgroup $T_{V}$, made of parallel translations, which acts on $V$ in a simply transitive way.

The action of $T_{V}$ defines even an absolute parallelism on $V$, i.e., it allows to canonically identify the tangent space $T_{v}V$ at an arbitrary point $v\in V$ of the affine space $V$, with the vector space $V$: in particular, if an origin $o\in V$ is chosen, then the differential

[TABLE]

of $g\in\operatorname{\mathsf{Aff}}(V)$ at $o$ can be regarded as an isomorphism of $V$, that is as an element of $\operatorname{\mathsf{GL}}(V)$. This explains the rightmost arrow of (2) and allows to regard $\operatorname{\mathsf{GL}}(V)$ as the linear group of the affine group, that is, as the subgroup $\operatorname{\mathsf{Aff}}(V)_{o}=\operatorname{\mathsf{GL}}(T_{o}V)$ of the group $\operatorname{\mathsf{Aff}}(V)$ that stabilizes the origin $o\in V$: this leads to the semidirect decomposition

[TABLE]

of the affine group $\operatorname{\mathsf{Aff}}(V)$, associated with the origin $o\in V$.

If now a subgroup $H\subset\operatorname{\mathsf{Aff}}(V)$ is given, decomposition (3) does not need to descend to $H$, in the sense that the sequence

[TABLE]

may be still exact, but not split. This remark motivates the following definition.

Definition 1.3**.**

We say that a subgroup $H\subset\operatorname{\mathsf{Aff}}(V)$ is of affine type if $H$ admits a semidirect decomposition

[TABLE]

for some $o\in V$. The subgroup $L_{H}=H_{o}$ is called the linear subgroup of $H$, whereas $T_{W}$ is its subgroup of translations.

The next proposition shows that condition (A2) is precisely about $\tau(H^{(k-1)})$ being a subgroup of affine type; it also shows that the symbol $W$ appearing above, in the case of a subgroup of affine type, corresponds to an actual linear subspace of $V$.

Proposition 1.1**.**

A subgroup $H\subset\operatorname{\mathsf{Aff}}(V)$ is of affine type if and only if an origin $o\in V$ can be chosen, such that:

• (i)

$W:=H\cdot o$* is a linear subspace, and*

• (ii)

$T(W)\subset H$.

Proof.

Suppose that there exists $o\in V$, such that (i) and (ii) hold. Let $h\in H$ be an arbitrary element: then $h\cdot o\in W$ implies that $T_{h\cdot o}\in T(W)\subset H$, whence

[TABLE]

where the first factor belongs to $T_{V}\cap H=T^{-1}(H)=T_{W}$, and the second clearly stabilizes $o$, that is, belongs to $H_{o}$.

Conversely, if $H$ is of affine type, then $W:=H\cdot o=(T_{V}\cap H)\cdot o=T^{-1}(H)\cdot o$ can be identified with the Abelian subgroup $T_{W}$ of $V$, that is, is a linear subspace (i); its image $T(W)$ in $\operatorname{\mathsf{Aff}}(V)$, being identified with $T_{V}\cap H$, is clearly contained in $H$ (ii). ∎

Remark 1.1**.**

Condition (ii) cannot be dropped: for example, the 1-parameter subgroup of “screw movements” along a line is not an affine type subgroup, but it has a line as an orbit.

Let $H=T_{W}\rtimes L_{H}$ be a subgroup of affine type, where $L_{H}=H_{o}$ denotes its linear subgroup, and let us fix a complementaty subspace $U$ to $W$ in $V$: then any $h\in H$ can be decomposed into a product

[TABLE]

Therefore, in terms of the decomposition $V=U+W$, the action of the linear part $L_{h}$ takes the form

[TABLE]

We let $\overline{L}_{H}:=\{\overline{L}_{h}\mid h\in H\}$.

Lemma 1.1**.**

Let $H\subset\operatorname{\mathsf{Aff}}(V)$ be a subgroup of affine type. Then there exists a 1–1 correspondence between$\overline{L}_{H}$–invariant hypersurfaces $\overline{\Sigma}\subset\overline{V}=V/W$ and (cylindrical) $H$–invariant hypersurfaces ${\Sigma}=W+\overline{\Sigma}$ in $V$.

Proof.

Let $\pi:V\to\overline{V}=V/W$ be the projection. Then if $\overline{\Sigma}$ is an $L_{H}$–invariant hypersureface in $\overline{V}$, then $\Sigma:=\pi^{-1}(\bar{\Sigma})$ is an $H$–invariant hypersurface in $V$, see also [1, Lemma 3.1]. ∎

1.2.2. $k$–admissible homogeneous manifolds

In what follows, unless otherwise specified, $o$ is a fixed point of $M=G/H$ (an “origin”), so that $M=G\cdot o$, and $o^{\ell}$ is a point of $J^{\ell}$ projecting onto $o$. This allows us to consider, $\forall\,\ell\geq 2$, the fibre $J^{\ell}_{o^{\ell-1}}$ as a vector space with the origin $o^{\ell}$ playing the role of zero vector. The group $G$ acts naturally on each $\ell$–jet space $J^{\ell}$:

[TABLE]

with $o\in S$, for all $g\in G$.

Definition 1.4**.**

The system $\mathcal{E}$ is called $G$–invariant if $G\cdot\mathcal{E}=\mathcal{E}$.

We denote by $H^{(\ell)}$ the stability subgroup $G_{o^{\ell}}$ in $G$ of the point $o^{\ell}$:

[TABLE]

We are going to assume that there exists a point $o^{k}\in J^{k}$, with $k\geq 2$, such that:

(A1) the orbit

[TABLE]

through the projection $o^{k-1}\in J^{k-1}$ of $o^{k}$ is open;

(A2) the orbit

[TABLE]

of the natural affine action

[TABLE]

in the fibre $J^{k}_{o^{k-1}}$ is an affine subspace and the group $\tau(H^{(k-1)})$ is a subgroup of affine type, i.e.,

[TABLE]

where $L_{H^{(k-1)}}$ is the stabilizer of $o^{k}$, see Definition 1.3.

Assumption (A2) implies that there is a point $o^{k}\in J^{k}_{o^{k-1}}$ such that the restriction of the affine bundle $\pi_{k,k-1}:J^{k}\to J^{k-1}$ to the orbit $G\cdot o^{k}$ is an affine subbundle of $\pi_{k,k-1}$ (over the base $\check{J}^{k-1}$).

The problem of classifying all $G$–invariant PDEs $\mathcal{E}\subset J^{k}$ on a given $(n+1)$–dimensional manifold $M$ acted upon by a Lie group $G$ will be made more workable by assuming $M$ to be a $G$–homogeneous manifold of a particular kind, namely a $k$–admissible one.

Definition 1.5**.**

A homogeneous manifold $M=G/H$ is called $k$–admissible for $k\geq 2$ if assumptions (A1) and (A2) are satisfied.

1.3. Natural bundles on jet spaces

1.3.1. The lift of hypersurfaces of $M$ to $J^{\ell}$

The space $J^{\ell}$ has a natural structure of smooth manifold: one way to see this is to extend the local coordinate system (1) on $M$ to a coordinate system

[TABLE]

on $J^{\ell}$, where each coordinate function222The $u_{i_{1}\cdots i_{k}}$’s are symmetric in the lower indices. $u_{i_{1}\cdots i_{k}}$, with $k\leq\ell$, is unambiguously defined by

[TABLE]

In formula (12) above the symbol $\partial_{i}$ denotes the partial derivative $\partial_{x^{i}}$, for $i=1,\ldots,n$; we recall that the hypersurface $S=S_{f}$ is the graph of the function $u=f(\boldsymbol{x})$ and, as such, it is admissible for the chart $(u,\boldsymbol{x})$.

The $\ell$–lift of $S$ is defined by

[TABLE]

It is an $n$–dimensional submanifold of $J^{\ell}$. If $S=S_{f}$ is the graph of $u=f(\boldsymbol{x})$, then $S_{f}^{(\ell)}$ can be naturally parametrized as follows:333We stress once again that a switch has occurred between the first and the second entry, with respect to a more standard literature.

[TABLE]

1.3.2. The tautological bundle and the higher order contact distribution on $J^{\ell}$

Lemma 1.2**.**

Any point $a^{\ell}=[S]^{\ell}_{\boldsymbol{p}}\in J^{\ell}$ canonically defines the $n$–dimensional subspace

[TABLE]

Definition 1.6**.**

The tautological rank–$n$ vector bundle $\mathcal{T}^{\ell}\subset\pi_{\ell,\ell-1}^{*}(TJ^{\ell-1})$ is the bundle over $J^{\ell}$ whose fiber over the point $a^{\ell}$ is given by (13), i.e.,

[TABLE]

The (truncated) total derivatives

[TABLE]

constitute a local basis of the bundle $\mathcal{T}^{\ell}$.

The pre–image $\mathcal{C}^{\ell}:=(d\pi_{\ell,\ell-1})^{-1}\mathcal{T}^{\ell}$ of the tautological bundle on $J^{\ell}$, via the differential $d\pi_{\ell,\ell-1}$ of the canonical projection $\pi_{\ell,\ell-1}$, is a distribution on $J^{\ell}$.

Definition 1.7**.**

$\mathcal{C}^{\ell}$ is called the $\ell^{\textrm{th}}$ order contact structure or the Cartan distribution (on $J^{\ell}$).

We will need also the vertical subbundle $T^{v}J^{\ell}:=\ker(d\pi_{\ell,\ell-1})$ of $TJ^{\ell}$. The distribution $\mathcal{C}^{\ell}$ has been called the “higher order contact structure” [6, 7] because, for $\ell=1$, if $(u,x^{i},u_{i})$ is a chart on $J^{1}$, then $\mathcal{C}:=\mathcal{C}^{1}=\ker(\theta)$, where $d\theta=du-u_{i}dx^{i}$, is the contact distribution.

1.3.3. The affine structure of the bundles $J^{\ell}\to J^{\ell-1}$ for $\ell\geq 2$

According to Definition 1.6, the tautological bundle $\mathcal{T}:=\mathcal{T}^{1}$ is the vector bundle over $J^{1}$ defined by

[TABLE]

Definition 1.8**.**

The normal bundle $\mathcal{N}$ is the line bundle

[TABLE]

over $J^{1}$.

Remark 1.2**.**

To simplify notations, we denote by $\partial_{u}$ the equivalence class $\partial_{u}\,\mathrm{mod}\,\mathcal{T}$.

Lemma 1.3 and Proposition 1.2 below are both well known (see for instance [6, 12]).

Lemma 1.3**.**

For $\ell\geq 1$, the following vector bundle isomorphism holds:

[TABLE]

Proposition 1.2**.**

For $\ell\geq 2$, the bundles $J^{\ell}\to J^{\ell-1}$ are affine bundles modeled by the vector bundles $\pi_{\ell-1,1}^{*}(S^{\ell}\mathcal{T}^{*}\otimes\mathcal{N})$. In particular, once a chart $(u,\boldsymbol{x})$ has been fixed, a choice of a point $[S]^{\ell}_{\boldsymbol{p}}$ (the origin) defines an identification of $J^{\ell}_{[S]^{\ell-1}_{\boldsymbol{p}}}$ with $S^{\ell}T^{*}_{\boldsymbol{p}}S$.

1.4. Constructing $G$–invariant PDEs $\mathcal{E}$

Let $M=G/H=G\cdot o$, $o\in M$, be an $(n+1)$–dimensional homogeneous manifold and recall (see Section 1.2.2) that $G$ acts on each jet space $J^{\ell}=J^{\ell}(n,M)$. To further simplify the setting, we will assume that $M=G/H$ possess a fiducial hypersurface of order $k$, defined below.

1.4.1. The fiducial hypersurface

Definition 1.9**.**

Let $S\subset M$ be a hypersurface, such that $S\ni o$. The hypersurface $S$ is called a a fiducial hypersurface (of order $k$), if $S$ is homogeneous with respect to a subgroup of $G$, such that (A1) and (A2) of Section 1.2.2 are satisfied with $o^{k}:=[S]^{k}_{o}$.

Plainly, if $M=G/H$ admits a fiducial hypersurface of order $k$, then it is $k$–admissible as well (see Definition 1.5). Let $S$ be a fiducial hypersurface of order $k$ in the sense of Definition 1.9: therefore, for any $\ell\leq k$, we will regard the point

[TABLE]

as the origin of $J^{\ell}$. Furthermore, the identification

[TABLE]

in the case when the fiducial hypersurface $S$ is the graph $S_{f}$ of a $f$, reads (see Proposition 1.2):

[TABLE]

We will use this identification in the sequel.

1.4.2. A general method for constructing $G$–invariant PDEs

We apply now Lemma 1.1 to the subgroup $\tau(H^{(k-1)})\subset\operatorname{\mathsf{Aff}}(J^{k}_{o^{k-1}})$ of affine type, which eventually leads to [1, Theorem 3.1].

Corollary 1.1**.**

Let $M=G/H$ be a $k$–admissible homogeneous manifold. Then there exists a $1$–$1$ correspondence between $L_{H^{(k-1)}}$–invariant hypersurfaces $\overline{\Sigma}\subset J^{k}_{o^{k-1}}/W^{k}$ and (cylindrical) $\tau(H^{(k-1)})$–invariant hypersurfaces $\Sigma=p^{-1}(\overline{\Sigma})\subset J^{k}_{o^{k-1}}$, where

[TABLE]

is the natural projection.

The aforementioned main result of [1], that is Theorem 3.1, is a direct consequence of above Corollary 1.1 and below Lemma 1.4, applied to the bundle $\pi_{k,k-1}$.

Lemma 1.4**.**

Let $\pi:P\longrightarrow B$ be a bundle. Assume that a Lie group $G$ of automorphisms of $\pi$, such that $B=G/H$, acts transitively on $B$, where $H$ is the stabilizer of a point $o\in B$. Then:

• i)

any $H$–invariant function $F$ on $P_{o}:=\pi^{-1}(o)$ extends to a $G$–invariant function $\widehat{F}$ on $P$ (where $\widehat{F}(gy)=F(y)$ for $y\in P_{o}$ and $g\in G$), and $F\longmapsto\widehat{F}$ is a bijection;

• ii)

*any $H$–invariant hypersurface $\Sigma$ of the fiber $P_{o}$ extends to a $G$–invariant hypersurface $\mathcal{E}_{\Sigma}:=G\cdot\Sigma$ of $P$, and this gives a bijection between $H$–invariant hypersurfaces of $P_{o}$ and $G$–invariant hypersurfaces of $P$. *

Proof.

See [1, Lemma 3.2]. ∎

Theorem 1.1**.**

Let $M=G/H$ be a $k$–admissible homogeneous manifold (see Definition 1.5). Then there is a natural $1$–$1$ correspondence between $L_{H^{(k-1)}}$– invariant hypersurfaces $\overline{\Sigma}$ (see also (10)) of $J^{k}_{o^{k-1}}/W^{k}$ and $G$–invariant hypersurfaces $\mathcal{E}_{\overline{\Sigma}}:=\mathcal{E}_{p^{-1}(\overline{\Sigma})}=G\cdot p^{-1}(\overline{\Sigma})$ of $J^{k}=J^{k}(n,M)$, where $p$ is the natural projection (16).

Above Theorem 1.1 closes the summary of the theory developed by the authors in [1], that is a strategy for constructing $G$–invariant PDEs imposed on the hypersurfaces of a $k$–admissible homogeneous manifold $M=G/H$:

1. (1)

calculate the orbit $W^{k}=\tau(H^{(k-1)})\cdot o^{k}$ and decompose $\tau(H^{(k-1)})$ accordingly to (10); 2. (2)

describe $L_{H^{(k-1)}}$–invariant hypersurfaces $\overline{\Sigma}\subset{V}^{k}=J^{k}_{o^{k-1}}/W^{k}$; 3. (3)

write down the $G$–invariant equations $\mathcal{E}_{\overline{\Sigma}}=G\cdot p^{-1}(\overline{\Sigma})$ in the coordinates (11).

In the next Section 2 we begin implementing this strategy for the projective space $\mathbb{P}^{n+1}$, whereas in Section 3 we will be dealing with the affine space $\mathbb{A}^{n+1}$; the $G$–invariant PDE itself is obtained, in an unified manner, in the Section 4.

2. Stabilizers of the $\operatorname{\mathsf{SL}}(n+2)$–action on $J^{\ell}(n,\mathbb{P}^{n+1})$

We consider the linear space $W:=\mathbb{R}^{n+2}$ with the basis

[TABLE]

and we let $G=\operatorname{\mathsf{SL}}(n+2)$ act naturally on it; therefore, $G$ acts on the projectivization $M:=\mathbb{P}W$ of $W$. The projective coordinates

[TABLE]

on $\mathbb{P}W=\mathbb{P}^{n+1}$ will be given by the dual coordinates to the basis above. We shall also need a scalar product

[TABLE]

on $E:=\left\langle e_{1},\ldots,e_{n}\right\rangle$, of signature $d,n-d$. The role of the fiducial hypersurfaces will be then played by the projective quadric

[TABLE]

where $W_{0}$ is the null cone of the quadratic form

[TABLE]

that is $W_{0}:=\{g_{W}=0\}$, and $\boldsymbol{x}=x^{i}e_{i}$ denotes the generic element of $E$, whose coordinates in the basis $\{e_{1}\,,e_{2}\,,\ldots\,,e_{n}\}$ are $(x^{1},\ldots,x^{n})$. The point

[TABLE]

clearly belongs to the fiducial hypersurface $\mathcal{S}_{g}$, so that it makes sense to consider

[TABLE]

for $k\geq 0\,.$ In particular, the point $o^{(1)}=[\mathcal{S}_{g}]^{1}_{o}$, that is the tangent space $T_{o}\mathcal{S}_{g}=T_{o}(\mathcal{S}_{g}\cap\mathcal{U})\in J^{1}$, in the affine coordinate neighborhood

[TABLE]

can be identified with $E=\ker d_{o}u$: indeed, $\ker d_{o}(tu-g)=\ker d_{o}u-d_{o}g=\ker d_{o}u=E$, because $d_{o}g=0$.

Lemma 2.1**.**

The stabilising subgroups corresponding to the origins $o^{(k)}$, for $k=0,1,2,3$, are:

[TABLE]

Proof.

An element of $G$ stabilizing the line generated by $p$ is a $(n+2)\times(n+2)$ matrix with determinant one, displaying all zeros in the first column, save for the first entry, that has to be equal to the inverse of the determinant of the rightmost lower $(n+1)\times(n+1)$ block: in other words,

[TABLE]

The same can be seen on the infinitesimal level: passing to the Lie algebra $\mathfrak{g}$ of $G$, we consider the decomposition

[TABLE]

where $\mathfrak{so}(W)=\mathfrak{so}(d+1,n+1-d)$ is identified with the space of skew–symmetric forms $\Lambda^{2}W$ and $S^{2}_{0}(W)$ denotes the space of trace–free symmetric forms with respect to $g_{W}$, cf. (19); therefore, since $W$ splits into the sum

[TABLE]

of the $(n+1)$–dimensional space $\mathbb{R}q\oplus E$ and the one–dimensional subspace $\mathbb{R}p$, we obtain the decompositions

[TABLE]

A quick remark about notation: by the symbol $\mathbb{R}p\wedge(E\oplus\mathbb{R}q)$ (resp., $\mathbb{R}p\odot(E\oplus\mathbb{R}q)$) we mean the image of the tensor product

[TABLE]

via the canonical projection of $W^{\otimes 2}$ onto $\Lambda^{2}W$ (resp. $S^{2}W$); since $\mathbb{R}p\cap(E\oplus\mathbb{R}q)=0$, we immediately see that both the aforementioned projections are isomorphisms, i.e., we can safely identify

[TABLE]

From (22) and (23)–(24) we obtain then

[TABLE]

which, in view of (25), becomes

[TABLE]

It is now easy to see that the Lie algebra $\mathfrak{g}_{0}=\mathfrak{h}$ of the stabilizer $H$ is nothing but

[TABLE]

embedded into $\mathfrak{g}$ via the diagonal embedding of the space $\mathbb{R}p\otimes(E\oplus\mathbb{R}q)$ into its square and the identity of $\mathfrak{gl}(E\oplus\mathbb{R}q)$. Thanks to the metric given by $g_{W}$ we can further identify

[TABLE]

thus obtaining

[TABLE]

The last identification allows to rewrite (21) as follows:

[TABLE]

where the factor $\operatorname{\mathsf{GL}}(T_{o}M)$ (resp., $T_{o}M$) is the image (resp., kernel) of the isotropy representation

[TABLE]

In light of what we have found it is easy to pass from (26) to the Lie algebra $\mathfrak{h}^{(1)}$ of $H^{(1)}$

[TABLE]

where

[TABLE]

On the level of Lie groups this means that

[TABLE]

or, more intrinsically,

[TABLE]

We shall show now that the subgroup of (30) that stabilizes $o^{(2)}$ is precisely

[TABLE]

where

[TABLE]

is the group of rigid motions of $E\simeq T_{o}\mathcal{S}_{g}$. To begin with, the very definition of the isotropy representation (cf. (27)) tells us that the factor $T_{o}M$ of $H^{(1)}$ survives in $H^{(2)}$; it is also easy to see that the “conformal factor” $\mathbb{R}^{\times}$, since it scales the dependent variable in a Darboux coordinate system, does not affect the second jet at zero of the quadric $\mathcal{S}_{g}$: the latter being $g$ itself, if we identify second–order jets with quadratic forms (see Section 1.3.3). Similarly, a transformation coming from the $\operatorname{\mathsf{GL}}(T_{o}\mathcal{S}_{g})$ component of $\operatorname{\mathsf{E}}(T_{o}\mathcal{S}_{g})$ preserves $o^{(2)}$ if and only if it preserves $g$: therefore, it must be an element of $\operatorname{\mathsf{O}}(T_{o}\mathcal{S}_{g})$.

In order to finish the proof of (31) it then remains to show that the “translational” component $T_{o}\mathcal{S}_{g}$ of $\operatorname{\mathsf{E}}(T_{o}\mathcal{S}_{g})$ does not move $o^{(2)}$: such a phenomenon can be better appreciated from a local coordinate perspective, which is better suited to the affine case; therefore we refer the reader to the proof of the analogous property in Section 3 below: see Remark 3.1. There, we will also prove that the aforementioned component does move $o^{(3)}$, eventually showing that

[TABLE]

thus concluding the whole proof. ∎

Remark 2.1**.**

The structure of $H^{(1)}$ is that of

[TABLE]

where

[TABLE]

with $\mathcal{H}$ being the $(2n+1)$–dimensional Heisenberg group. Indeed, from (28) and (29) it follows that

[TABLE]

that is,

[TABLE]

In terms of traceless $(n+2)\times(n+2)$ matrices, an element

[TABLE]

of the algebra (32) corresponds to the matrix

[TABLE]

Remark 2.2**.**

The fiducial hypersurface $\mathcal{S}_{g}$ is a homogenous manifold: namely,

[TABLE]

It is indeed convenient, before passing to the application of Theorem 1.1, to prove the analogous result for the affine case: after that, the two cases will go on in parallel, due to the fact that the structure of the model fiber of $J^{3}$ over $J^{2}$ does not feel the topology of the underlying manifold, that has changed from $\mathbb{P}^{n+1}$ to $\mathbb{A}^{n+1}$.

3. Stabilizers of the $\operatorname{\mathsf{Aff}}(n+1)$–action on $J^{\ell}(n,\mathbb{A}^{n+1})$

By the symbol $\mathbb{A}^{n+1}$ we denote the linear space $\mathbb{R}^{n+1}$, regarded as an affine space (over itself). As a group acting transitively on $\mathbb{A}^{n+1}$ we take

[TABLE]

The coordinates on $\mathbb{R}^{n+1}$ are now going to be

[TABLE]

dual to the basis

[TABLE]

i.e., we have the same $n$–dimensional space $E$ as before, with the same coordinates, but now the $(n+1)$–dimensional underlying manifold is

[TABLE]

Since $\mathbb{A}^{n+1}$ still possesses the zero, we set $o:=0\in\mathbb{A}^{n+1}$. The role of a fiducial hypersurface, however, will be played now by a quadric

[TABLE]

where $g$ is the same scalar product on $E$ as before, see (17), and it coincides with (19) in the affine neighborhood (20). As before, we let $o^{(k)}:=[\mathcal{S}_{g}^{\textrm{aff}}]_{o}^{k}$, for $k\geq 0$, so that the point $o^{(1)}=[\mathcal{S}_{g}^{\textrm{aff}}]_{o}^{1}$ will be again the tangent space $T_{o}\mathcal{S}_{g}^{\textrm{aff}}\in J^{1}$, that is the hyperplane $E=\mathbb{R}^{n}$ of $\mathbb{R}^{n+1}$.

Lemma 3.1**.**

The stabilizing subgroups of the origins $o^{(k)}$, for $k=0,1,2,3$, are:

[TABLE]

Proof.

Formula (34) is well–known: an affine transformation preserves the zero if and only if it is linear.

Concerning (35), let us note that a $(n+1)\times(n+1)$ non–singular matrix preserves the hyperplane $\mathbb{R}^{n}$ if and only if it has the form

[TABLE]

where $A\in\operatorname{\mathsf{GL}}(n)$, $\boldsymbol{w}\in\mathbb{R}^{n}$ and $\mu\in\mathbb{R}^{\times}$. In what follows it is going to be useful to separate the following three factors (beware the abuse of notation):

[TABLE]

Let us observe that:

• •

the matrix $A$ acts naturally on the hyperplane $\mathbb{R}^{n}$, while not affecting the complementary line $\left\langle e_{0}\right\rangle$;

• •

the scalar $\mu$ rescales the elements of the line $\left\langle e_{0}\right\rangle$, while not affecting the complementary hyperplane $\mathbb{R}^{n}$;

• •

the vector $\boldsymbol{w}$ acts on the affine hyperplane $u=1$ by translation: in particular, it “tilts” the line $\left\langle e_{0}\right\rangle$ into the line $\left\langle e_{0}+\boldsymbol{w}\right\rangle$.

This will be crucial to understand how these transformations affect the fiducial hypersurface $\mathcal{S}_{g}^{\textrm{aff}}$.

Now let us finish the proof that $\operatorname{Stab}_{H}(o^{(1)})=(\mathbb{R}^{n}\rtimes\operatorname{\mathsf{GL}}(n))\times\mathbb{R}^{\times}$. To this end, it suffices to observe that

[TABLE]

whence it immediately follows that $\operatorname{Stab}_{H}(o^{(1)})$ is the direct product of the subgroup of matrices

[TABLE]

and the multiplicative group $\mathbb{R}^{\times}$. It remains to observe that the generic element of $\mathbb{R}^{n}\rtimes\operatorname{\mathsf{GL}}(n)$ corresponds to a matrix of the form (37).

Let us pass to (36), i.e., to the computation of the stabilizer $\operatorname{Stab}_{H^{(1)}}(o^{(2)})$ of the second jet at $0\in\mathbb{R}^{n}$ of the quadric hypersurface $\mathcal{S}_{g}^{\textrm{aff}}$. In view of the structure of $H^{(1)}$, we may treat each transformation $A$, $\boldsymbol{w}$ and $\mu$ separately. The crucial remark is that, thanks to the identification

[TABLE]

the point $o^{(2)}$ is identified with $2g\otimes\partial_{u}$, where $g$ is the scalar product (17), cf. (33), see also Remark 1.2. In particular,

[TABLE]

therefore $A\cdot o^{(2)}=o^{(2)}$ if and only if $A^{*}(g)=g$, that is, $A\in\operatorname{\mathsf{O}}(d,n-d)$. It then remains to prove that both transformations $w$ and $\mu$ do not move $o^{(2)}$.

It is easy to see that the transformation $\mu$ maps $\partial_{u}$ into $\mu^{-1}\partial_{u}$ while simultaneously transforming $g$ into $\mu g$, where the symbol $\mu^{-1}$ denotes the inverse of the non–zero number $\mu$. In view of (38), this means that

[TABLE]

Transformations of type $\boldsymbol{w}$ take a more radical toll on the fiducial hypersurface $\mathcal{S}_{g}^{\textrm{aff}}$. However, even if the resulting hypersurface $\boldsymbol{w}(\mathcal{S}_{g}^{\textrm{aff}})$ looks like a “slanted paraboloid” (see the picture below), the second–order jet at zero of $\boldsymbol{w}(\mathcal{S}_{g}^{\textrm{aff}})$ will be the same as the original hypersurface $\mathcal{S}_{g}^{\textrm{aff}}$.

In order to see this, let

[TABLE]

be the quadratic form associated to the scalar product (17). Note that from $\mathcal{S}_{g}^{\textrm{aff}}=\{(Q(\boldsymbol{x}),\boldsymbol{x})\mid\boldsymbol{x}\in\mathbb{R}^{n}\}$ we get

[TABLE]

Observe that the function $\boldsymbol{t}(\boldsymbol{x}):=\boldsymbol{x}+Q(\boldsymbol{x})\boldsymbol{w}$ is a small deformation of the identity in a sufficiently small neighborhood of zero. As such, $\boldsymbol{t}(\boldsymbol{x})$ will admit a (local) inverse. We claim, that

[TABLE]

approximates the inverse of $\boldsymbol{t}(\boldsymbol{x})$ up to third–order terms. Indeed,

[TABLE]

This will allow us to work with the graph of the function $f(\boldsymbol{t}):=Q(\boldsymbol{x}(\boldsymbol{t}))$ instead of the hypersurface $\boldsymbol{w}(\mathcal{S}_{g}^{\textrm{aff}})$, as long as only jets from zero up to second order are concerned. In particular,

[TABLE]

since the Jacobian $\frac{\partial\boldsymbol{x}}{\partial\boldsymbol{t}}$ at zero is the identity. We have then proved that $[f]_{0}^{1}=o^{(1)}$. Analogously,

[TABLE]

since the first derivatives of $Q$ vanish at the origin. Then we also have that $[f]_{0}^{2}=o^{(2)}$. This concludes the proof that $H^{(2)}=(\mathbb{R}^{n}\rtimes\operatorname{\mathsf{O}}(d,n-d))\times\mathbb{R}^{\times}$: indeed only the second factor of $H^{(1)}$ has become smaller.

In order to deal with the last case, i.e., $H^{(3)}$, it suffices to observe that, in view of the third–order analogue of formula (38), viz.

[TABLE]

transformations of type $A$ (where now $A\in\operatorname{\mathsf{O}}(d,n-d)$) and of type $\mu$ do not change the third–order jet at zero $o^{(3)}$ of the fiducial hypersurface $\mathcal{S}_{g}^{\textrm{aff}}$. However, the third–order jet at zero of $\boldsymbol{w}(\mathcal{S}_{g}^{\textrm{aff}})$ will not be the same as $o^{(3)}$, unless $\boldsymbol{w}=0$. We have already observed that $f(\boldsymbol{t})$ and $Q(\boldsymbol{x})$ have the same derivatives at 0 up to order two.

To study the third–order jet at zero of $\boldsymbol{w}(\mathcal{S}_{g}^{\textrm{aff}})$ we need compute the third derivatives of $f$, where now $f(\boldsymbol{t})=Q(\boldsymbol{x}(\boldsymbol{t}))$, with $\boldsymbol{x}(\boldsymbol{t})$ being the true inverse of $\boldsymbol{t}(\boldsymbol{x})$, and not the approximated one, i.e., (39). The reason why we use the same symbol for both the exact and the approximated (local) inverse, beside an evident notation simplification, is that the final result will depend only on the approximated one.

[TABLE]

Evaluating the last expression at 0 we obtain

[TABLE]

Now, for the purpose of computing the second derivatives of $\boldsymbol{x}$ at 0 in (41), we can use the approximated inverse, that is (39):

[TABLE]

Indeed, the discrepancy between the true and the approximated inverse, being of third order in $\boldsymbol{x}$, will still vanish in 0, even after a double differentiation.

Therefore, the third–order term of the Taylor expansion of $f$ around 0 (where, it is worth stressing, $f$ is the one computed via the true inverse of $\boldsymbol{t}(\boldsymbol{x})$) is precisely

[TABLE]

Since we have already observed that $\boldsymbol{w}(\mathcal{S}_{g}^{\textrm{aff}})$ and $\mathcal{S}_{g}^{\textrm{aff}}$ have the same jets at 0 up to order 2, and the third–order derivatives of $Q$ are zero, formula (42) shows that $[\mathcal{S}_{g}^{\textrm{aff}}]_{0}^{3}=o^{(3)}$ if and only if $\boldsymbol{w}=0$.

This shows that $\operatorname{Stab}_{H^{(2)}}(o^{(3)})=\operatorname{\mathsf{O}}(d,n-d)\times\mathbb{R}^{\times}$, thus concluding the entire proof. ∎

Remark 3.1**.**

As we have anticipated, the proof of Lemma 3.1 above also provides the missing steps in the proof of Lemma 2.1; observe also that the residual action of the group $G$ on the fiber $J^{3}_{o^{2}}$ is exactly the same, that is, that of $\operatorname{\mathsf{E}}(\mathbb{R}^{n})\times\mathbb{R}^{\times}$. It is then reasonable to continue analyzing the two cases in parallel.

4. $\operatorname{\mathsf{PGL}}(n)$– and $\operatorname{\mathsf{Aff}}(n)$–invariant PDEs on hypersurfaces of $\mathbb{P}^{n+1}$ and $\mathbb{A}^{n+1}$

Proposition 4.1**.**

The projective hyperquadric $\mathcal{S}_{g}$ defined by (18) (resp. the quadric hypersurface $\mathcal{S}_{g}^{\textrm{aff}}$ defined by (33)) is a fiducial hypersurface of order both 2 and 3 with respect to the action of the affine group $\operatorname{\mathsf{Aff}}(n+1)$ on the affine space $\mathbb{A}^{n+1}$ (resp., of the projective group $\operatorname{\mathsf{SL}}(n+2)$ on the projective space $\mathbb{P}^{n+1}$), in the sense of Definition 1.5.

Proof.

The first part is obvious: see also the analogous [1, Proposition 4.1] in the Euclidean case. Indeed, $J^{1}$ is the same as $\mathbb{P}T^{*}\mathbb{R}^{n+1}$ or, equivalently, the flag space $F_{0,n}$, on which the linear group $\operatorname{\mathsf{GL}}(n+1)$ already acts transitively, let alone $\operatorname{\mathsf{Aff}}(n+1)$. So, $o^{(1)}$ is the flag $(0,\mathbb{R}^{n})$ and the action of $H$ on $J^{1}_{0}=\mathbb{P}(\mathbb{R}^{n+1\,\ast})$ is transitive. Therefore, since the $\operatorname{\mathsf{Aff}}(n+1)$–orbit of $o$ is the entire $M$, the $\operatorname{\mathsf{Aff}}(n+1)$–orbit of $o^{(1)}$ is the entire space $J^{1}$, viz.

[TABLE]

Let us now study the orbit $H^{(1)}\cdot o^{(2)}$ in $J^{2}_{o^{(1)}}$, bearing in mind the identification (38) and the description (35) of $H^{(1)}$. Since the quadratic form $Q$ associated to the scalar product is non–degenerate, its $\operatorname{\mathsf{GL}}(n)$–orbit will be open. Incidentally, we see the appearance of a $\operatorname{\mathsf{Aff}}(n+1)$–invariant second–order PDE, namely the Monge–Ampère equation $\mathcal{E}\subset J^{2}$ given by $\det(u_{ij})=0$.

Summing up,

[TABLE]

is an open subset of $J^{2}(n,\mathbb{A}^{n+1})$ (which is contained in the complement $J^{2}(n,\mathbb{A}^{n+1})\smallsetminus\mathcal{E}$ of the Monge–Ampère equation $\mathcal{E}$). Therefore, the assumption (A1) of Definition 1.5 is met.

It remains to check assumption (A2) of Definition 1.5: we begin by showing that the orbit $H^{(2)}\cdot o^{(3)}$ is a proper affine sub–space of $J^{3}_{o^{(2)}}$. To this end, we shall need the identification (40). Indeed, from the proof of Lemma 3.1 above it is clear that the $H^{(2)}$–orbit of $o^{(3)}$ is made of the elements

[TABLE]

with $\boldsymbol{w}\in\mathbb{R}^{n}$. Therefore, from formula (42) it follows immediately that

[TABLE]

and then (40) allows to identify the difference $[\boldsymbol{w}(\mathcal{S}_{g}^{\textrm{aff}})]_{o}^{3}-o^{(3)}$ with the element

[TABLE]

of the vector space $S^{3}\mathbb{R}^{n\,\ast}\otimes\left\langle\partial_{u}\right\rangle$, where $\boldsymbol{w}^{\#}$ is the dual covector to $\boldsymbol{w}$ by means of the scalar product (17). In other words, as $\boldsymbol{w}$ ranges in $\mathbb{R}^{n}$, (43) describes the linear subspace

[TABLE]

By construction, this is the linear space modeling the fibre $H^{(2)}\cdot o^{(3)}$. Since the same is true for any fibre, assumption (A2) of Definition 1.5 is met; indeed, as we pointed out in Section 1.2.2 above, assumption (A2), in the case when (10) holds, is the same as having a (proper) affine sub–bundle, and (10) immediately follows from (36).

The projective case can be dealt with analogously. ∎

4.1. The main result

Theorem 4.1**.**

Fix a scalar product $g$ of signature $(d,n-d)$ as in (17) and let $\mathcal{S}_{g}^{\textrm{aff}}\subset\mathbb{A}^{n+1}$ (resp., $\mathcal{S}_{g}\subset\mathbb{P}^{n+1}$) be the corresponding fiducial (quadratic) hypersurface. Let

[TABLE]

denote the space of trace–free cubic forms on $\mathbb{R}^{n}$. Then, for any $\operatorname{\mathsf{CO}}(d,n-d)$–invariant hypersurface

[TABLE]

we obtain an $\operatorname{\mathsf{Aff}}(n+1)$–invariant third–order PDE $\mathcal{E}_{\Sigma}\subset J^{3}(n,\mathbb{A}^{n+1})$ (resp., an $\operatorname{\mathsf{SL}}(n+2)$–invariant third–order PDE $\mathcal{E}_{\Sigma}\subset J^{3}(n,\mathbb{P}^{n+1})$).

Proof.

Let us begin with the affine case. The first step consists in proving that $\tau_{\mathcal{R}}(H^{(2)})$–invariant hypersurfaces in

[TABLE]

are the same as $\operatorname{\mathsf{CO}}(p,n-p)$–invariant hypersurfaces in $S_{0}^{3}\mathbb{R}^{n\,\ast}$. To this end, recall the structure of $H^{(2)}$, studied in Lemma 3.1 (see, in particular, formula (36)) and observe that the factor $\mathbb{R}^{\times}$ acts by multiplication by $\mu\in\mathbb{R}^{\times}$ on $N_{o}\mathcal{S}_{g}^{\textrm{aff}}$. The factor $\operatorname{\mathsf{O}}(p,n-p)$ acts naturally on $S^{3}T_{o}^{*}\mathcal{S}_{g}^{\textrm{aff}}$, which can be identified with $S^{3}\mathbb{R}^{n\,\ast}$. According to Proposition 4.1 above, an element $\boldsymbol{w}$ in the factor $\mathbb{R}^{n}$ acts by shifting along $R_{o^{(2)}}=\mathbb{R}^{n\,\ast}\odot\left\langle g\right\rangle$ by $-2\boldsymbol{w}^{\#}\odot g$, see also (43), and hence its action on the quotient is trivial.

The claim then follows from Theorem 1.1, recalling that $\operatorname{\mathsf{CO}}(d,n-d)=\operatorname{\mathsf{O}}(d,n-d)\times\mathbb{R}^{\times}$.

Since the projective case can be dealt with analogously, we omit the proof. ∎

4.2. Coordinate description

Since the problem is, by its nature, a local one, we shall not consider the projective case, since the affine space $\mathbb{A}^{n+1}$ can be considered as an affine neighborhood embedded in $\mathbb{P}^{n+1}$. Again, we extend the global coordinate system $\{u,x^{1},\ldots,x^{n}\}$ of $\mathbb{A}^{n+1}$ to a (local) coordinate system of $J^{3}(n,\mathbb{A}^{n+1})$: see also Section 1.3.1.

Lemma 4.1**.**

Let $\mathcal{E}_{\Sigma}$ be the $\operatorname{\mathsf{Aff}}(n+1)$–invariant equation associated to the $\operatorname{\mathsf{CO}}(d,n-d)$–invariant hypersurface $\Sigma$, as in Theorem 4.1 above. Then, in the aforementioned coordinate system on $J^{3}$, the equation $\mathcal{E}_{\Sigma}$ can be described as $\{f=0\}$, where the function $f=f(u_{ij},u_{ijk})$, that does not depend on $u,x^{1},\ldots,x^{n},u_{1},\ldots,u_{n}$, is the same function describing the hypersurface $\Sigma_{o^{(1)}}$ of $J^{3}_{o^{(1)}}$.

Proof.

It is a consequence of Lemma 1.4, where the bundle is

[TABLE]

and the subgroup $T\subset G=\operatorname{\mathsf{Aff}}(n+1)$ will be the $(2n+1)$–dimensional group

[TABLE]

The first factor of $T$ acts by translations on $\mathbb{R}^{n+1}$ and the lifted translations fix the $u_{i}$’s and, similarly, the $u_{ij}$’s and the $u_{ijk}$’s. Therefore, it is enough the first factor of $T$ to fulfil the hypothesis of Lemma 1.4.

Let us consider now

[TABLE]

Easy computations show that $\phi^{(1)\,\ast}(u_{i})=u_{i}+w_{i}$, whereas $\phi^{(2)\,\ast}(u_{ij})=u_{ij}$ and $\phi^{(3)\,\ast}(u_{ijk})=u_{ijk}$. The first fact shows that $T$ acts transitively on $J^{1}$ (since the translations act transitively on $J^{0}$ and the $\phi$’s act transitively on the fibres of $J^{1}\to J^{0}$). The second fact shows that $T$ acts trivially on the fibre $J^{3}_{o^{(1)}}$. Thus, the result follows from Lemma 1.4 applied to the group $T$. ∎

Example 4.1**.**

For $n=2$, a straightforward computation based on the proof of Lemma 4.1 (see [9, Section 6] for more details), shows that the subset $\mathcal{E}:=\{f=0\}$ of $J^{3}$, where

[TABLE]

is invariant with respect to the group $\operatorname{\mathsf{Aff}}(3)$. In [9] it is also shown that the same subset $\mathcal{E}$, in the real case, shows two different characters, depending on whether it projects over the open subset $\det u_{ij}>0$, or $\det u_{ij}<0$: the former corresponds to the invariant PDE associated with $\operatorname{\mathsf{CO}}(2)=\operatorname{\mathsf{CO}}(0,2)$, the latter to the invariant PDE associated with $\operatorname{\mathsf{CO}}(1,1)$; see also Section 4.4 below. In the first case, the invariant PDE is actually a system of two PDEs: this corresponds to (45) being the sum of two positive quantities; in the second case, the invariant subset $\mathcal{E}$ turns out to be the union of two scalar PDEs.

4.3. Complex $\operatorname{\mathsf{CO}}_{n}$-invariant hypersurfaces in $S^{3}_{0}(\mathbb{C}^{n})$, with $n=3,4$

The departing point of the main Theorem 4.1 is a $\operatorname{\mathsf{CO}}(d,n-d)$–invariant hypersurface $\Sigma$ in the trace–free third symmetric power $S_{0}^{3}\mathbb{R}^{n\,\ast}$ of the $n$–dimensional real vector space $\mathbb{R}^{n\,\ast}$. While a general classification in the real case is still unattainable, much can be said in the case of small values of $n$, if we work over the field of complex number.

Therefore, only in this section, $V=\mathbb{C}^{n}$ is going to be a complex vector space, with $n=3,4$: having set $W:=S^{3}_{0}(V)$, we shall study complex $\operatorname{\mathsf{CO}}(V)$–invariant hypersurfaces $\Sigma$ in $W$; in particular, there will be no signature, so that we consider the complex conformal group $\operatorname{\mathsf{CO}}(V)=\operatorname{\mathsf{CO}}_{n}(\mathbb{C})$, rather that its split real counterparts $\operatorname{\mathsf{CO}}_{d,n-d}(\mathbb{R})$.

More accurately, we will derive a description of complex invariant hypersurfaces $\Sigma$ in the irreducible $\operatorname{\mathsf{CO}}(V)$–module $W=S^{3}_{0}(V)$ of traceless symmetric 3–forms of the standard module $V=\mathbb{C}^{n}$ for $n=3$ and, partially, for $n=4$ from the known results of invariants’ theory, see [11, 13]; afterwards, one can reduce the description of the real hypersurfaces that are invariant with respect to the corresponding normal real forms $\operatorname{\mathsf{CO}}_{1,2}(\mathbb{R})$ and $\operatorname{\mathsf{SO}}_{2,2}(\mathbb{R})$, as well as with respect to the compact real forms $\operatorname{\mathsf{CO}}_{3}(\mathbb{R})$ and $\operatorname{\mathsf{CO}}_{4}(\mathbb{R})$, to the description of the real forms of the above–obtained complex hypersurfaces.

By employing the same notation of [10], we will denote by $R(k\pi_{1})$ the irreducible representation of the simple Lie algebra $\mathfrak{so}_{n}(\mathbb{C})$, whose highest weight is $k\pi_{1}$, always assuming that $n\geq 3$ and denoting by $\pi_{1}$ the first fundamental weight of $\mathfrak{so}_{n}(\mathbb{C})$: in particular, $R(\pi_{1})$ is the tautological representation in the space $V=\mathbb{C}^{n}$ and $R(3\pi_{1})$ is the highest irreducible component $W=S^{3}_{0}V$ in the symmetric cube $S^{3}V$.

4.3.1. The complex case with $n=3$

Recall that the Lie algebra $\mathfrak{so}_{3}(\mathbb{C})$ is isomorphic to the Lie algebra $\mathfrak{sl}(U)=\mathfrak{sl}_{2}(\mathbb{C})$ and that all irreducible $\mathfrak{sl}_{2}(\mathbb{C})$–modules are exhausted by the symmetric power $S^{k}U$ of the tautological module $U=\mathbb{C}^{2}$. The tensor product $S^{k}U\otimes S^{\ell}U^{\prime}$ is decomposed into irreducible submodules by the Klebsh–Gordon formula

[TABLE]

The tautological representation of $\mathfrak{so}_{3}(\mathbb{C})=\mathfrak{sl}_{2}(\mathbb{C})$ is the adjoint representation $V=S^{2}U$ and the representation

[TABLE]

This is the $\mathfrak{sl}_{2}(\mathbb{C})$-module of binary forms of order $6$. The full algebra of (polynomial) invariants $\mathbb{C}[S^{6}(U)]^{\mathfrak{sl}_{2}(\mathbb{C})}$ is known, see [11]. It is generated by 5 invariants $f_{2},f_{4},f_{6},f_{10},f_{15}$, of degrees 2, 4, 6, 10, 15, where the last invariant $f_{15}\in A:=\mathbb{C}[f_{2},f_{4},f_{6},f_{10}]$ and the algebra $A$ is the algebra of polynomials in four (independent) variables $f_{i}$.

Theorem 4.2**.**

Any complex $\operatorname{\mathsf{SO}}(V)$–invariant hypersurface in $S^{3}_{0}V$, with $V=\mathbb{C}^{3}$ has the form $\Sigma^{c}_{f}=\{f=c\}$ where $f\in\mathbb{C}[f_{2},f_{4},f_{6},f_{10}f_{15}]$ and $c\in\mathbb{C}$ is a constant. Any $\operatorname{\mathsf{CO}}(V)$-invariant hypersurface has the form $\Sigma^{0}_{f}=\{f=0\}$ where $f=f(f_{2},f_{4},f_{6},f_{10},f_{15})$ is a homogeneous polynomial of $f_{i}$, $\deg(f_{i})=i$.

Moreover, any homogeneous invariant hypersurface of degree $\leq 210$ has the form $f=0$ where the polynomial $f$ is given in Table 1; the explicit form of the generators can be found in [13].

4.3.2. The complex case with $n=4$

Consider now the case $n=4$. Then $\mathfrak{so}(V)=\mathfrak{so}_{4}(\mathbb{C})=\mathfrak{so}(U)+\mathfrak{so}(U^{\prime}),\,U=U^{\prime}=\mathbb{C}^{2}$ and the tautological module is $V=U\otimes U^{\prime}$. Then $S^{2}_{0}V=S^{2}U\otimes S^{2}U^{\prime}$ and

[TABLE]

Then $S_{0}^{3}V=S^{3}U\otimes S^{3}U^{\prime}$.

It is known that the algebra of invariants of the $\mathfrak{sl}_{2}(\mathbb{C})$-module $S^{3}U$ of ternary forms is generated by the discriminant $\delta$, see [13], where for

[TABLE]

the discriminant is

[TABLE]

Hence the algebra of invariants $\mathbb{C}[S^{3}_{0}V]^{\mathfrak{so}_{3}(\mathbb{C})}$ contains $\delta$ and $\delta^{\prime}$.

Theorem 4.3**.**

Any polynomial $f=f(\delta,\delta^{\prime})$ defines an invariant hypersurface $f=c$, where $c\in\mathbb{C}$ is constant. Any homogeneous polynomial $f=f(\delta,\delta^{\prime})$ defines an $\operatorname{\mathsf{CO}}(V)$ invariant hypersurface $f=0$.

We stress that not all invariants are polynomials of $\delta$ and $\delta^{\prime}$, that is, there may be other invariants.

4.3.3. A glimpse into the real case

A standard method to cook out real invariants, having at one’s disposal the complex ones, is by means of the anti–involution $\sigma$ in $\mathfrak{so}(\mathbb{C}^{n})$, i.e., the complex conjugation in $\mathbb{C}^{n}$: the anti–involution $\sigma$ determines the real form $\mathfrak{so}(k,\ell)$ and then the real and the imaginary parts of the complex generators of the algebra $A=\mathbb{C}[\mathbb{C}^{n}]^{\mathfrak{so}(\mathbb{C}^{n})}$ turn out to be real invariants that generate the whole real algebra of invariants. But there is a catch: the so–obtained real generators might by dependent.

Even though, in general, the description of a minimal system of generators of $A$ is a very complicated problem, in practice it is possible to describe the invariants in small degrees $k$.

The so–called symbolic method for constructing invariants boils down to obtaining scalar invariants by contracting tensor products $y_{1}\otimes y_{2}\otimes\cdots\otimes y_{k}$ of cubic forms with the inverse metric $g^{ij}$. For example, for $k=2$, one can construct the invariant $I=y_{ijk}z_{i^{\prime}j^{\prime}k^{\prime}}g^{ii^{\prime}}g^{jj^{\prime}}g^{kk^{\prime}}$; in the case of binary form, one has to use also the determinant $\det y_{ij}$.

This way, one can get a description of the invariants in small degree.

4.4. The $\operatorname{\mathsf{Aff}}(3)$ case

Going back to the real–differentiable setting, if we set $n=2$, then it is is easy to use the results contained into Theorem 4.1 and Lemma 4.1 above to write down explicitly the unique $\operatorname{\mathsf{Aff}}(3)$–invariant scalar third–order PDE $\mathcal{E}$ imposed on hypersurfaces of $\mathbb{A}^{3}$. To clarify what we mean by “unique”, it should be stressed from the outset that, in general, the $\operatorname{\mathsf{Aff}}(n+1)$–invariant PDE $\mathcal{E}\subset J^{3}$ constructed according to Theorem 4.1 projects onto an open subset $\check{J}^{2}$ of $J^{2}$: this is a direct consequence of the assumption (A1) on the action of $G$, see above Section 1.2.2. In turn, there are as many open subsets $\check{J}^{2}$, as the $\operatorname{\mathsf{GL}}(n)$–equivalence classes of fiducial hypersurfaces (33): if we denote by $d_{n}$ the ceiling of $n/2$, then these classes are labeled by the signatures

[TABLE]

i.e., there is $d_{n}+1$ of them. The union of all the subsets $\check{J}^{2}$ is dense in $J^{2}$ and its boundary is the unique $2^{\textrm{nd}}$ order $\operatorname{\mathsf{Aff}}(n+1)$–invariant PDE, that is the Monge–Ampère equation $\det\operatorname{hess}(u)=0$: see also the proof of the assumption (A1) of Proposition 4.1.

In the case $n=2$, we have only two open subsets of $J^{2}$, corresponding to the Riemanian $(+,+)$ and to the Lorenzian $(+,-)$ signature of the Hessian of the surface in $\mathbb{A}^{3}$, denoted respectively by $\check{J}^{2}_{+}$ and $\check{J}^{2}_{-}$. In view of the important link between the $\operatorname{\mathsf{Aff}}(3)$–invariant PDEs and the geometry of affine surfaces, below we sketch the relation between such PDEs and the Fubini-Pick invariant.

Let $u=f(x^{1},\dots,x^{n})$ describe a hypersurface $S$ of $\mathbb{A}^{n+1}$ which is the graph of the function $f$. Let us consider the basis

[TABLE]

The above basis is unimodular as $\det(\partial_{u}\,,\,D^{(1)}_{1}\,,\dots,D^{(1)}_{n})=1$. The components of the Blaschke metric $G$ are

[TABLE]

where

[TABLE]

whereas the component of the Fubini-Pick cubic form $C$ are

[TABLE]

where $D_{h}$ are the total derivatives, see also (14). The Fubini-Pick invariant is the function defined as

[TABLE]

which, in the case $n=2$ and up to a non-zero factor, is equal to the right-hand side term of (45). Then the $\operatorname{\mathsf{Aff}}(3)$–invariant PDE is $\mathcal{E}:=\{f=0\}$, with $f$ given by (45), see [9] for more details.

Another approach, based on the study of the singularities of the group action, that has been used in [8], lead to the very same equation (45).

We stress that the equation $\mathcal{E}$ projects onto the whole of $J^{2}$, because (45) is defined on the whole $J^{3}$: however, if we take the intersections

[TABLE]

we obtain precisely the two equations, say, $\mathcal{E}_{\Sigma_{+}}$ and $\mathcal{E}_{\Sigma_{-}}$, that come from Theorem 4.1: they correspond to the $\operatorname{\mathsf{CO}}(2)$–invariant subset $\Sigma_{+}:=\{0\}$ and to the $\operatorname{\mathsf{CO}}(1,1)$–invariant subset $\Sigma_{-}$ made of two invariant lines, respectively. In other words,

[TABLE]

whence the adjective “unique”.