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Hopf hypersurfaces in complex Grassmannians of rank two

Ruenn-Huah Lee, Tee-How Loo

Abstract

In this paper, we study real hypersurfaces in complex Grassmannians of rank two. First, the nonexistence of mixed foliate real hypersurfaces is proven. With this result, we show that for Hopf hypersurfaces in complex Grassmannians of rank two, the Reeb principal curvature is constant along integral curves of the Reeb vector field. As a result the classification of contact real hypersurfaces is obtained. We also introduce the notion of $q$-umbilical real hypersurfaces in complex Grassmannians of rank two and obtain a classification of such real hypersurfaces.

Hopf hypersurfaces in complex Grassmannians of rank two

Abstract

In this paper, we study real hypersurfaces in complex Grassmannians of rank two. First, the nonexistence of mixed foliate real hypersurfaces is proven. With this result, we show that for Hopf hypersurfaces in complex Grassmannians of rank two, the Reeb principal curvature is constant along integral curves of the Reeb vector field. As a result the classification of contact real hypersurfaces is obtained. We also introduce the notion of -umbilical real hypersurfaces in complex Grassmannians of rank two and obtain a classification of such real hypersurfaces.

Paper Structure

This paper contains 6 sections, 35 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. The complex Grassmannians of rank two
  3. Real hypersurfaces in $\hat{M}^m(c)$
  4. Hopf hypersurfaces in $\hat{M}^m(c)$
  5. Mixed foliate real hypersurfaces in $\hat{M}^m(c)$
  6. $q$-umbilical real hypersurfaces in $\hat{M}^m(c)$

Key Result

Lemma 3.1

If $\xi\in\mathfrak D$, then $A\phi\xi_a=0$ for $a\in\{1,2,3\}$.

Theorems & Definitions (57)

  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 47 more