On Umbilical Real Hypersufaces of Products of Complex Space Forms

TL;DR

This paper investigates the existence and classification of totally umbilical real hypersurfaces in products of complex space forms, extending known results and establishing new rigidity and nonexistence theorems.

Contribution

It provides new rigidity and nonexistence results for totally umbilical hypersurfaces in product spaces of complex space forms, including a classification under certain conditions.

Findings

No totally umbilical hypersurfaces without local product structure exist in these spaces.

Hypersurfaces with local almost product structure are either totally geodesic or extrinsic hyperspheres.

Shape operator cannot be parallel for hypersurfaces lacking local product structure.

Abstract

Tashiro and Tachibana proved that there exist no totally umbilical hypersurfaces in complex space forms with nonzero constant holomorphic sectional curvature, and it is also known that the shape operator of such hypersurfaces cannot be parallel. Motivated by these results, we study real hypersurfaces in products of complex space forms. We establish rigidity and nonexistence results for totally umbilical real hypersurfaces in this setting. In particular, we show that if a real hypersurface in a product of complex space forms does not admit a local product structure, then its shape operator cannot be parallel. Moreover, we provide a classification of totally umbilical real hypersurfaces, showing that those admitting a local almost product structure are necessarily totally geodesic or extrinsic hyperspheres.