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On Umbilical Real Hypersufaces of Products of Complex Space Forms

Iury Domingos, Ranilze da Silva, Alexandre de Sousa, Feliciano Vitório

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.

On Umbilical Real Hypersufaces of Products of Complex Space Forms

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.

Paper Structure

This paper contains 7 sections, 11 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Complex manifolds
  4. Real hypersurfaces in products of complex space forms
  5. Products of complex space forms
  6. Real hypersurfaces and their fundamental equations
  7. Umbilical hypersurfaces in products of complex space forms

Key Result

Proposition 3.1

The curvature tensor $\overline{\mathcal{R}} \colon T\overline{M} \times T\overline{M} \times T\overline{M} \rightarrow T\overline{M}$ of $\overline{M} = \mathbb{CQ}_1 \times \mathbb{CQ}_2$ is given by where $\overline{X},\overline{Y},\overline{Z} \in T\overline{M}$ and $\overline{L}_i = I + \varepsilon_i F$, with $\varepsilon_1 = 1$ and $\varepsilon_2 = -1$.

Theorems & Definitions (20)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 4.1: Tashiro--Tachibana & Niegerball--Ryan
  • Proposition 4.2
  • proof
  • ...and 10 more