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Projectively and affinely invariant PDEs on hypersurfaces

Dmitri V. Alekseevsky, Gianni Manno, Giovanni Moreno

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.

Projectively and affinely invariant PDEs on 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 -invariant PDEs imposed on hypersurfaces of an -dimensional homogeneous space , under mild assumptions on the Lie groups . In the present paper the method is applied to the case when or and the homogeneous space is the -dimensional projective or affine space, respectively. The paper's main result is that projectively or affinely invariant PDEs with independent and one unknown variables are in one-to-one correspondence with -invariant hypersurfaces of the space of trace-free cubic forms in variables. Local descriptions are also provided.

Paper Structure

This paper contains 14 sections, 9 theorems, 79 equations.

Table of Contents

  1. A general construction of $G$--invariant PDEs on a homogeneous manifold $M = G/H$
  2. Preliminary definitions
  3. Assumptions on the Lie group $G$
  4. Affine groups and their subgroups of affine type
  5. $k$--admissible homogeneous manifolds
  6. Natural bundles on jet spaces
  7. The lift of hypersurfaces of $M$ to $J^\ell$
  8. The tautological bundle and the higher order contact distribution on $J^{\ell}$
  9. The affine structure of the bundles $J^{\ell}\to J^{\ell-1}$ for $\ell\geq 2$
  10. Constructing $G$--invariant PDEs $\mathcal{E}$
  11. The fiducial hypersurface
  12. A general method for constructing $G$--invariant PDEs
  13. Stabilizers of the $\mathop{\mathrm{\mathsf{SL}}}\nolimits(n+2)$--action on $J^\ell(n,\mathbb{P}^{n+1})$
  14. Stabilizers of the $\mathop{\mathrm{\mathsf{Aff}}}\nolimits(n+1)$--action on $J^\ell(n,\mathbb{A}^{n+1})$

Key Result

Lemma 1.1

Let $H \subset \mathop{\mathrm{\mathsf{Aff}}}\nolimits(V)$ 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{\

Theorems & Definitions (25)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 1.1
  • proof
  • Definition 1.4
  • Definition 1.5
  • Lemma 1.2
  • Definition 1.6
  • Definition 1.7
  • ...and 15 more