Affine hypersurfaces of arbitrary signature with an almost symplectic form

TL;DR

This paper classifies affine hypersurfaces with arbitrary signature that possess an almost symplectic form, showing that higher order parallelism conditions restrict the shape operator's rank, extending previous Lorentzian results.

Contribution

It provides a complete classification of affine hypersurfaces with higher order parallel almost symplectic forms, generalizing prior Lorentzian hypersurface classifications.

Findings

If $R^p ext{or} abla^p ext{omega}=0$, the shape operator's rank is at most one.

The results extend classifications to hypersurfaces with arbitrary signature.

Provides a comprehensive understanding of affine hypersurfaces with higher order parallel almost symplectic forms.

Abstract

In this paper we study affine hypersurfaces with non-degenerate second fundamental form of arbitrary signature additionally equipped with an almost symplectic structure $\omega$. We prove that if $R^p\omega=0$ or $\nabla^p\omega =0$ for some positive integer $p$ then the rank of the shape operator is at most one. The results provide complete classification of affine hypersurfaces with higher order parallel almost symplectic forms and are generalization of recently obtained results for Lorentzian affine hypersurfaces.