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Cone Conditions for the Curvature Operator of the Second Kind on Einstein Manifolds

Haiqing Cheng, Kui Wang

TL;DR

The paper investigates sphere theorems for closed Einstein manifolds under a cone condition on the curvature operator of the second kind $\mathring{R}$. By establishing a Bochner-type inequality $\langle \Delta R, R\rangle \ge 0$ under the cone constraint $\mathcal{C}(\alpha,\theta)$ with real $\alpha$ in $[1, (n+2)(n-1)/2)$ and $\theta>-1$, the authors show that such manifolds are forced to be flat or round spheres; equality cases yield rigid spectral structures. The results extend prior integer-$\alpha$ cases to real $\alpha$ and provide explicit $\theta(n,\alpha)$ parameters, broadening the known rigidity phenomena for Einstein manifolds with curvature operator of the second kind in cone conditions. This contributes a unified framework for sphere theorems via cone conditions, with potential implications for rigidity classifications in Riemannian geometry.

Abstract

In this note, we study Einstein manifolds whose curvature operator of the second kind $\mathring{R}$ satisfies the cone condition \[ α^{-1}\big(\sum_{i=1}^{[α]} λ_i+ (α- [α] ) λ_{[α] + 1} \big) \ge -θ\barλ \] for some real number $α\in [1, (n+2)(n-1)/2)$. Here $[α] :=\max\{ m \in \mathbb{Z}: m \leq α\}$, $θ>-1$ and $λ_1 \le \cdots \le λ_{(n+2)(n-1)/2}$ are the eigenvalues of $\mathring{R}$ and $\barλ$ is their average. The main result states that any closed Einstein manifold of dimension $n \ge 4$ with $\mathring{R}$ satisfies the cone condition is flat or a round sphere. These results generalize recent works corresponding to $α\in \mathbb Z_+$ of the authors \cite{CW24-1,CW25-2} and Fu-Lu \cite{FL25}.

Cone Conditions for the Curvature Operator of the Second Kind on Einstein Manifolds

TL;DR

The paper investigates sphere theorems for closed Einstein manifolds under a cone condition on the curvature operator of the second kind . By establishing a Bochner-type inequality under the cone constraint with real in and , the authors show that such manifolds are forced to be flat or round spheres; equality cases yield rigid spectral structures. The results extend prior integer- cases to real and provide explicit parameters, broadening the known rigidity phenomena for Einstein manifolds with curvature operator of the second kind in cone conditions. This contributes a unified framework for sphere theorems via cone conditions, with potential implications for rigidity classifications in Riemannian geometry.

Abstract

In this note, we study Einstein manifolds whose curvature operator of the second kind satisfies the cone condition \[ α^{-1}\big(\sum_{i=1}^{[α]} λ_i+ (α- [α] ) λ_{[α] + 1} \big) \ge -θ\barλ \] for some real number . Here , and are the eigenvalues of and is their average. The main result states that any closed Einstein manifold of dimension with satisfies the cone condition is flat or a round sphere. These results generalize recent works corresponding to of the authors \cite{CW24-1,CW25-2} and Fu-Lu \cite{FL25}.
Paper Structure (5 sections, 6 theorems, 54 equations)

This paper contains 5 sections, 6 theorems, 54 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Curvature Operator of the Second Kind
  4. A formula for Weyl tensor on Einstein manifolds
  5. Proof of Theorems \ref{['thm1']} and \ref{['thm2']}

Key Result

Theorem 1.1

Let $(M^n, g)$ be a closed Einstein manifold of dimension $n \ge 6$, and let $\mathring{R}$ be the curvature operator of the second kind. If $\mathring{R} \in \mathcal{C}(\alpha,\theta(n,\alpha) )$ for $1 \le \alpha \le \min\left\{\frac{n^4-n^3+8 n-8}{3 n^3+5 n^2-22 n+8}, \frac{n^2+n-8}{4n-8} \right then $M$ is flat or a round sphere. Here $N=\frac{(n+2)(n-1)}{2}$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.1
  • Definition 2.1: NPW22
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 3 more