On $3$-dimensional $\widetilde{J}$-tangent centro-affine hypersurfaces and $\widetilde{J}$-tangent affine hyperspheres with some null-directions

TL;DR

This paper classifies 3-dimensional centro-affine hypersurfaces and affine hyperspheres in  with a para-complex structure, focusing on null-directions of the second fundamental form and providing explicit examples.

Contribution

It provides a complete local classification of -dimensional  hypersurfaces with -structure and null-directions, identifying conditions under which they are affine hyperspheres and hyperquadrics.

Findings

Hypersurfaces with two null-directions are affine hyperspheres and hyperquadrics.

Explicit examples of such hypersurfaces are constructed.

Classification results extend understanding of -structure in affine differential geometry.

Abstract

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^4$. In this paper we study $3$-dimensional centro-affine hypersurfaces with a $\widetilde{J}$-tangent centro-affine vector field (sometimes called $\widetilde{J}$-tangent centro-affine hypersurfaces) as well as $3$-dimensional $\widetilde{J}$-tangent affine hyperspheres with the property that at least one null-direction of the second fundamental form coincides with either $\mathcal{D}^+$ or $\mathcal{D}^-$. The main purpose of this paper is to give a full local classification of the above mentioned hypersurfaces. In particular, we prove that every nondegenerate centro-affine hypersurface of dimension $3$ with a $\widetilde{J}$-tangent centro-affine vector field which has two null-directions $\mathcal{D}^+$ and $\mathcal{D}^-$ must be both an affine hypersphere and a hyperquadric. Some examples of these hypersurfaces are also given.