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

Abstract

Let $\tilde{J}$ be the canonical para-complex structure on ${\mathbb{R}}^4$. In this paper we study $3$-dimensional centro-affine hypersurfaces with a $\tilde{J}$-tangent centro-affine vector field (sometimes called $\tilde{J}$-tangent centro-affine hypersurfaces) as well as $3$-dimensional $\tilde{J}$-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 $\tilde{J}$-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.

arXiv admin note: text overlap with arXiv:1804.01599