Let $\Jt$ be the canonical para-complex structure on $\R^4$. In this paper we study $3$-dimensional centro-affine hypersurfaces with a $\Jt$-tangent centro-affine vector field (sometimes called $\Jt$-tangent centro-affine hypersurfaces) as well as $3$-dimensional $\Jt$-tangent affine hyperspheres with the property that at least one null-direction of the second fundamental form coincides with either $\DD^+$ or $\DD^-$. 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 $\Jt$-tangent centro-affine vector field that has two null-directions $\DD^+$ and $\DD^-$ must be both an affine hypersphere and a hyperquadric. Some examples of these hypersurfaces are also given.
"On $3$-dimensional $\Jt$-tangent centro-affine hypersurfaces and $\Jt$-tangent affine hyperspheres with some null-directions,"
Turkish Journal of Mathematics: Vol. 42:
5, Article 54.
Available at: https://journals.tubitak.gov.tr/math/vol42/iss5/54