Parallel almost paracontact structures on affine hypersurfaces
Volume 122 / 2019
Abstract
Let $\widetilde{J}$ be the canonical paracomplex structure on $\mathbb{R}^{2n+2}\simeq\widetilde{\mathbb{C}}^{n+1}$. We study real affine hypersurfaces $f\colon M\rightarrow \widetilde{\mathbb{C}}^{n+1}$ with a $\widetilde{J}$-tangent transversal vector field. Such a vector field induces in a natural way an almost paracontact structure $(\varphi,\xi,\eta)$ on $M$ as well as an affine connection $\nabla$. In this paper we give a classification of hypersurfaces with the property that $\varphi$ or $\eta$ is parallel relative to the connection $\nabla$. Moreover, we show that if $\nabla\varphi=0$ (respectively $\nabla\eta=0$) then around each point of $M$ there exists a parallel almost paracontact structure. We illustrate the results with appropriate examples.
