Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ricci pinched compact hypersurfaces in spheres

Marcos Dajczer, Miguel I. Jimenez, Theodoros Vlachos

Abstract

We investigate the topology of the compact hypersurfaces in round spheres whose Ricci curvature satisfies an appropriate bound that only depends on the mean curvature of the submanifold. In this paper, the use of the Bochner technique allows same stronger results than the ones obtained by us in the case of submanifolds lying in any codimension.

Ricci pinched compact hypersurfaces in spheres

Abstract

We investigate the topology of the compact hypersurfaces in round spheres whose Ricci curvature satisfies an appropriate bound that only depends on the mean curvature of the submanifold. In this paper, the use of the Bochner technique allows same stronger results than the ones obtained by us in the case of submanifolds lying in any codimension.
Paper Structure (3 sections, 8 theorems, 51 equations)

This paper contains 3 sections, 8 theorems, 51 equations.

Table of Contents

  1. The Bochner operator
  2. A parametrization
  3. The proofs

Key Result

Theorem 1

Let $f\colon M^n\to\mathbb{S}^{n+1}$, $n\geq 4$, be an isometric immersion of a compact manifold. Assume that $f$ satisfies the pinching condition $(*)$ for some $k\geq 2$ where $k<n/2$ if $n$ is even and $k<(n-1)/2$ if $n$ is odd. Then $M^n$ is simply connected, hence orientable, and one of the fol and $H_{n-k-1}(M^n;\mathbb{Z})=\mathbb{Z}^{\beta_{k+1}(M)}$, where $\beta_{k+1}(M)$ denotes the $(k

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Lemma 5
  • Proposition 6
  • Proposition 7
  • Theorem 8