$\tilde{J}$-tangent affine hypersurfaces with an induced almost paracontact structure

Abstract

We study real affine hypersurfaces $f:M\to {\mathbb{R}}^{2n+2}$ with an almost paracontact structure $(\phi ,\xi ,\eta )$ induced by a $\tilde{J}$-tangent transversal vector filed, where $\tilde{J}$ is the canonical paracomplex structure on ${\mathbb{R}}^{2n+2}$. We give a classification of hypersurfaces for which an induced almost paracontact structure is metric relative to the second fundamental form. Some other properties of such hypersurfaces are also studied.