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New Curvature Conditions for the Bochner Technique

Peter Petersen, Matthias Wink

Abstract

We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti number of closed $n$-dimensional Riemannian manifolds with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in dimensions $n=5,6$ we obtain vanishing of the Betti numbers provided that the curvature operator is $3$-positive. As Böhm-Wilking observed, $3$-positivity of the curvature operator is not preserved by the Ricci flow.

New Curvature Conditions for the Bochner Technique

Abstract

We prove a vanishing and estimation theorem for the -Betti number of closed -dimensional Riemannian manifolds with a lower bound on the average of the lowest eigenvalues of the curvature operator. This generalizes results due to D. Meyer, Gallot-Meyer, and Gallot. For example, in dimensions we obtain vanishing of the Betti numbers provided that the curvature operator is -positive. As Böhm-Wilking observed, -positivity of the curvature operator is not preserved by the Ricci flow.

Paper Structure

This paper contains 7 sections, 21 theorems, 113 equations.

Table of Contents

  1. Preliminaries
  2. Tensors
  3. The regular representation
  4. The Bochner Technique
  5. Controlling the curvature term of Lichnerowicz Laplacians
  6. Geometric Applications
  7. Examples

Key Result

Theorem A

Let $n \geq 3$ and let $(M,g)$ be a closed connected $n$-dimensional Riemannian manifold. Fix $1 \leq p \leq \lfloor \frac{n}{2} \rfloor$ and consider the eigenvalues $\lambda_1 \leq \ldots \leq \lambda_{ {\binom{n}{2}} }$ of the curvature operator of $(M,g).$ If $\lambda_1 + \ldots + \lambda_{n-p} In case $\kappa =0$ all harmonic $p$-forms are parallel. When $\kappa \leq 0$ and $\operatorname{di

Theorems & Definitions (60)

  • Theorem A
  • Corollary
  • Example
  • Remark
  • Corollary
  • Theorem B
  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • ...and 50 more