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Manifolds with harmonic curvature and curvature operator of the second kind

Haiping Fu, Yao Lu, Zhilin Dai

TL;DR

The paper investigates compact $n\ge3$-dimensional manifolds with harmonic curvature under nonnegativity conditions on the curvature operator of the second kind. It combines a Bochner-type analysis for the traceless Ricci tensor with the $\mathrm{NP}$ weighted-sum calculus to derive lower bounds on $\langle \Delta E,E\rangle$ dictated by $\mathring{R}$, proving the manifold is Einstein under the stated bound. Depending on $n$, additional rigidity results imply constant sectional curvature: $3\le n\le7$ under $\frac{n(n+2)}{2(n+1)}$-nonnegativity, $8\le n\le13$ under $4$-nonnegativity, and $n\ge14$ under $[\frac{n+2}{4}]$-nonnegativity. These results extend prior work of Kashiwada, Dai–Fu, and Li, highlighting a strong link between harmonic curvature, curvature-operator positivity, and global curvature rigidity.

Abstract

We prove that compact Riemannian manifolds of dimension $n\ge3$ with harmonic curvature and $\frac{n(n+2)}{2(n+1)}$-nonnegative curvature operator of the second kind must be Einstein. In particular, Building upon Dai-Fu's work \cite{DF}, it follows that if the curvature operator of the second kind is $\min\{\frac{n(n+2)}{2(n+1)},\max\{4,\frac{(n+2)}{4}\}\}$-nonnegative , then such a manifold must be of constant curvature.

Manifolds with harmonic curvature and curvature operator of the second kind

TL;DR

The paper investigates compact -dimensional manifolds with harmonic curvature under nonnegativity conditions on the curvature operator of the second kind. It combines a Bochner-type analysis for the traceless Ricci tensor with the weighted-sum calculus to derive lower bounds on dictated by , proving the manifold is Einstein under the stated bound. Depending on , additional rigidity results imply constant sectional curvature: under -nonnegativity, under -nonnegativity, and under -nonnegativity. These results extend prior work of Kashiwada, Dai–Fu, and Li, highlighting a strong link between harmonic curvature, curvature-operator positivity, and global curvature rigidity.

Abstract

We prove that compact Riemannian manifolds of dimension with harmonic curvature and -nonnegative curvature operator of the second kind must be Einstein. In particular, Building upon Dai-Fu's work \cite{DF}, it follows that if the curvature operator of the second kind is -nonnegative , then such a manifold must be of constant curvature.
Paper Structure (2 sections, 4 theorems, 17 equations)

This paper contains 2 sections, 4 theorems, 17 equations.

Table of Contents

  1. introduction
  2. proof of Theorem \ref{['A']}

Key Result

Theorem 1.1

Let $(M, g)$ be an $n(\ge 3)$-dimensional compact Riemannian manifold with harmonic curvature. If the curvature operator of the second kind $\mathring{R}$ is $\frac{n(n+2)}{2(n+1)}$-nonnegative, then $M$ is an Einstein manifold.

Theorems & Definitions (8)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • proof : Proof of Theorem \ref{['A']}