Einstein manifolds under cone conditions for the curvature operator of the second kind

TL;DR

This paper extends classification results of Einstein manifolds by relaxing curvature conditions to a cone condition, proving that under this weaker assumption, the manifold must be flat or a sphere in specific dimensions.

Contribution

It introduces a new cone condition on the curvature operator of the second kind and proves classification results for Einstein manifolds in certain dimensions under this condition.

Findings

Manifolds are flat or spherical under the cone curvature condition in dimensions 4, 5, and ≥8.

The cone condition is strictly weaker than two-nonnegative curvature operator.

The results generalize previous classifications with less restrictive curvature assumptions.

Abstract

It is established in [6, 14, 23] that any closed Einstein manifold with two-nonnegative curvature operator of the second kind is either flat or a round sphere. In this paper, we refine this result by relaxing the curvature condition to a cone condition (strictly weaker than two nonnegativity) proposed by Li [18]. Precisely, we prove that any closed Einstein manifold of dimension $n=4$ or $n=5$ or $n\ge 8$, if the curvature operator of the second kind $\mathring{R}$ satisfies \begin{align*} (\lambda_1+\lambda_2)/2 \ge -\theta(n) \bar \lambda, \end{align*} then the manifold is either flat or a round sphere. Here, $\lambda_1\le \lambda_2\le \cdots\le \lambda_{(n-1)(n+2)/2}$ are the eigenvalues of $\mathring{R}$, $ \bar \lambda $ is their average, and $\theta(n)$ is a positive constant defined as in (1.2).