# On $4$-dimensional Lorentzian affine hypersurfaces with an almost symplectic form

## TL;DR

This paper classifies 4-dimensional Lorentzian affine hypersurfaces with an almost symplectic form, showing the shape operator's rank is at most one under certain parallelism conditions, advancing understanding of their geometric structure.

## Contribution

It provides a classification result for Lorentzian affine hypersurfaces with higher order parallel almost symplectic forms, focusing on the rank of the shape operator.

## Key findings

- Rank of the shape operator is at most one under specific conditions.
- Complete classification of hypersurfaces with higher order parallel almost symplectic forms.
- Establishes conditions for the vanishing of certain curvature-related operators.

## Abstract

In this paper we study $4$-dimensional affine hypersurfaces with a Lorentzian second fundamental form additionally equipped with an almost symplectic structure $\omega$. We prove that the rank of the shape operator is at most one if $R^k\cdot \omega=0$ or $\nabla^k\omega=0$ for some positive integer $k$. This result is the final step in a classification of Lorentzian affine hypersurfaces with higher order parallel almost symplectic forms.