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Einstein submanifolds with parallel mean curvature into $\mathbb{S}^n\times\mathbb{R}$

Estela Garcia, Fernando Manfio

Abstract

We prove that Einstein submanifolds in $\mathbb{S}^n\times\mathbb{R}$ with flat normal bundle and parallel mean curvature are warped product of isometric immersions. Key words: Einstein submanifolds, Parallel mean curvature, Flat normal bundle.

Einstein submanifolds with parallel mean curvature into $\mathbb{S}^n\times\mathbb{R}$

Abstract

We prove that Einstein submanifolds in with flat normal bundle and parallel mean curvature are warped product of isometric immersions. Key words: Einstein submanifolds, Parallel mean curvature, Flat normal bundle.
Paper Structure (6 sections, 13 theorems, 88 equations)

This paper contains 6 sections, 13 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic facts on submanifolds in $\mathbb{Q}^n_\epsilon\times\mathbb{R}$
  4. Submanifolds with flat normal bundle in $\mathbb{Q}^n_\epsilon\times\mathbb{R}$
  5. A basic result
  6. The proof of the main result

Key Result

Theorem 1.1

Let $f\colon M^m\to\mathbb{Q}^n_\epsilon\times\mathbb{R}$, $m\geq3$, be an isometric immersion of a connected Einstein manifold, with Ricci curvature $\lambda$ and flat normal bundle. Then $f$ belongs to class $\mathcal{A}$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • Proposition 4.4
  • proof
  • Remark 4.5
  • Lemma 5.1
  • ...and 17 more