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Manifolds with harmonic Weyl curvature and curvature operator of the second kind

Haiping Fu, Yao Lu

TL;DR

The paper addresses rigidity questions for compact Riemannian manifolds with harmonic Weyl curvature under lower bounds on the curvature operator of the second kind $\mathring{R}$. By deriving a refined Bochner formula and employing a weighted-sum framework for the eigenvalues of $\mathring{R}$, the authors prove that, under sharp nonnegativity thresholds, such manifolds must be globally conformally equivalent to spaces of positive constant curvature or flat. They establish three main results: (i) a high-dimensional Tachibana-type rigidity (Theorem A) for $n\ge 8$ with $\frac{3(n-1)(n+2)}{4(3n-1)}$-nonnegativity; (ii) a second set of rigidity statements (Theorem B) for $n\ge 8$ with $2$-nonnegative (and $n\ge 11$ with $3$-nonnegative) curvature operator of the second kind, plus a 5- and 7-dimensional case with explicit constants; and (iii) a complete four-dimensional classification under a cone-type bound on the eigenvalues, including cases conformal to $\mathbb{CP}^2$ with Fubini–Study metric or products of constant-curvature surfaces. These results extend and unify prior Tachibana-type rigidity results for curvature operators of the second kind, providing near-optimal thresholds in several dimensions and a definitive four-dimensional classification under the stated cone condition.

Abstract

We prove that a compact Riemannian manifold of dimension $n\ge 8$ with harmonic Weyl curvature and $\frac{3(n-1)(n+2)}{4(3n-1)}$-nonnegative curvature operator of the second kind is either globally conformally equivalent to a space of positive constant curvature or is isometric to a flat manifold. In particular, We also give a classification of four-dimensional manifolds with harmonic Weyl curvature satisfying a cone condition. This result generalizes the work in \cite{DFY24,FLD,Li22}.

Manifolds with harmonic Weyl curvature and curvature operator of the second kind

TL;DR

The paper addresses rigidity questions for compact Riemannian manifolds with harmonic Weyl curvature under lower bounds on the curvature operator of the second kind . By deriving a refined Bochner formula and employing a weighted-sum framework for the eigenvalues of , the authors prove that, under sharp nonnegativity thresholds, such manifolds must be globally conformally equivalent to spaces of positive constant curvature or flat. They establish three main results: (i) a high-dimensional Tachibana-type rigidity (Theorem A) for with -nonnegativity; (ii) a second set of rigidity statements (Theorem B) for with -nonnegative (and with -nonnegative) curvature operator of the second kind, plus a 5- and 7-dimensional case with explicit constants; and (iii) a complete four-dimensional classification under a cone-type bound on the eigenvalues, including cases conformal to with Fubini–Study metric or products of constant-curvature surfaces. These results extend and unify prior Tachibana-type rigidity results for curvature operators of the second kind, providing near-optimal thresholds in several dimensions and a definitive four-dimensional classification under the stated cone condition.

Abstract

We prove that a compact Riemannian manifold of dimension with harmonic Weyl curvature and -nonnegative curvature operator of the second kind is either globally conformally equivalent to a space of positive constant curvature or is isometric to a flat manifold. In particular, We also give a classification of four-dimensional manifolds with harmonic Weyl curvature satisfying a cone condition. This result generalizes the work in \cite{DFY24,FLD,Li22}.
Paper Structure (7 sections, 13 theorems, 66 equations)

This paper contains 7 sections, 13 theorems, 66 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Notion and Lemmas
  4. Weighted principle
  5. Bochner formula for manifold with harmonic Weyl curvature
  6. The case for $n=5$ and $n=7$
  7. Rigidity theorems for dimensional four

Key Result

Theorem A

Let $(M, g)$ be an $n(\ge 8)$-dimensional compact Riemannian manifold with harmonic Weyl curvature tensor. If the curvature operator of the second kind $\mathring{R}$ is $\frac{3(n-1)(n+2)}{4(3n-1)}$-nonnegative, then $M$ is either globally conformally equivalent to a space of positive constant curv

Theorems & Definitions (30)

  • Theorem A
  • Corollary 1.1
  • Theorem B
  • Remark 1.2
  • Theorem C
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 20 more