- Turkish Journal of Mathematics
- Vol: 42 Issue: 5
- On $3$-dimensional $\Jt$-tangent centro-affine hypersurfaces and $\Jt$-tangent affine hyperspheres w...
On $3$-dimensional $\Jt$-tangent centro-affine hypersurfaces and $\Jt$-tangent affine hyperspheres with some null-directions
Authors : Zuzanna Szancer
Pages : 2779-2797
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Publication Date : 9999-12-31
Article Type : Makaleler
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.Keywords : Centro-affine hypersurface, almost paracontact structure, affine hypersphere, null-direction