Turkish Journal of Mathematics
Abstract
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.
DOI
10.3906/mat-1804-27
Keywords
Centro-affine hypersurface, almost paracontact structure, affine hypersphere, null-direction
First Page
2779
Last Page
2797
Recommended Citation
SZANCER, Z (2018). On $3$-dimensional $\Jt$-tangent centro-affine hypersurfaces and $\Jt$-tangent affine hyperspheres with some null-directions. Turkish Journal of Mathematics 42 (5): 2779-2797. https://doi.org/10.3906/mat-1804-27
