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Tachibana-type Theorems and special Holonomy

Peter Petersen, Matthias Wink

Abstract

We prove rigidity results for compact Riemannian manifolds in the spirit of Tachibana. For example, we observe that manifolds with divergence free Weyl tensors and $\lfloor \frac{n-1}{2} \rfloor$-nonnegative curvature operators are locally symmetric or conformally equivalent to a quotient of the sphere. The main focus of the paper is to prove similar results for manifolds with special holonomy. In particular, we consider Kähler manifolds with divergence free Bochner tensor. For quaternion Kähler manifolds we obtain a partial result towards the LeBrun-Salamon conjecture.

Tachibana-type Theorems and special Holonomy

Abstract

We prove rigidity results for compact Riemannian manifolds in the spirit of Tachibana. For example, we observe that manifolds with divergence free Weyl tensors and -nonnegative curvature operators are locally symmetric or conformally equivalent to a quotient of the sphere. The main focus of the paper is to prove similar results for manifolds with special holonomy. In particular, we consider Kähler manifolds with divergence free Bochner tensor. For quaternion Kähler manifolds we obtain a partial result towards the LeBrun-Salamon conjecture.

Paper Structure

This paper contains 6 sections, 18 theorems, 94 equations.

Table of Contents

  1. Preliminaries
  2. Tensors
  3. The Bochner technique
  4. Manifolds with divergence free Weyl tensors
  5. Kähler manifolds with divergence free Bochner tensors
  6. Quaternion Kähler manifolds

Key Result

Theorem A

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold with divergence free Weyl tensor. If $(M,g)$ has $\lfloor \frac{n-1}{2} \rfloor$-nonnegative curvature operator, then $(M,g)$ is locally symmetric or conformally equivalent to a quotient of the standard sphere.

Theorems & Definitions (36)

  • Theorem A
  • Theorem B
  • Theorem C
  • Corollary
  • Example 1.1
  • Remark 1.2
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • Corollary 1.5
  • ...and 26 more