Parallel Almost Paracontact Structures on Affine Hypersurfaces

TL;DR

This paper classifies affine hypersurfaces in para-complex space with parallel almost paracontact structures, revealing conditions under which these structures are globally parallel, supported by examples.

Contribution

It provides a classification of hypersurfaces with parallel almost paracontact structures induced by a para-complex structure, including conditions for local parallelism.

Findings

Hypersurfaces with $ abla ext{} ext{varphi}=0$ or $ abla ext{} ext{eta}=0$ are locally equipped with parallel structures.

Explicit examples illustrating the classification and parallel structures.

Conditions for global parallelism of almost paracontact structures on affine hypersurfaces.

Abstract

Let $\widetilde{J}$ be the canonical para-complex 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 vector field induces in a natural way an almost paracontact structure ${(\varphi,\xi,\eta)}$ on $M$ as well as the affine connection $\nabla$. In this paper we give the 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. Results we illustrate with some examples.