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Parallel almost paracontact structures on affine hypersurfaces

Volume 122 / 2019

Zuzanna Szancer Annales Polonici Mathematici 122 (2019), 249-265 MSC: Primary 53A15; Secondary 53D15. DOI: 10.4064/ap180527-6-12 Published online: 17 May 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.

Authors

  • Zuzanna SzancerDepartment of Applied Mathematics
    University of Agriculture in Kraków
    253c Balicka Street
    30-198 Kraków, Poland
    e-mail

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