Parallel Almost Paracontact Structures on Affine Hypersurfaces

Abstract

Let $\tilde{J}$ be the canonical para-complex structure on ${\mathbb{R}}^{2n+2}\simeq {\tilde{\mathbb{C}}}^{n+1}$. We study real affine hypersurfaces $f:M\to {\tilde{\mathbb{C}}}^{n+1}$ with a $\tilde{J}$-tangent transversal vector field. Such vector field induces in a natural way an almost paracontact structure $(\phi ,\xi ,\eta )$ on $M$ as well as the affine connection $\nabla$. In this paper we give the classification of hypersurfaces with the property that $\phi$ or $\eta$ is parallel relative to the connection $\nabla$. Moreover, we show that if $\nabla \phi =0$ (respectively $\nabla \eta =0$) then around each point of $M$ there exists a parallel almost paracontact structure. Results we illustrate with some examples.

arXiv admin note: text overlap with arXiv:1804.02396, arXiv:1804.01599