Einstein manifolds of negative lower bounds on curvature operator of the second Kind

TL;DR

This paper proves that closed Einstein manifolds with a certain negative lower bound on the curvature operator of the second kind are either flat or spherical, refining previous results on nonnegative curvature operators.

Contribution

It establishes a new rigidity result for Einstein manifolds with negative bounds on the curvature operator of the second kind, improving earlier nonnegative curvature bounds.

Findings

Manifolds are either flat or spherical under the given curvature bounds.

The result generalizes and strengthens previous nonnegative curvature operator theorems.

Provides explicit bounds involving eigenvalues of the curvature operator.

Abstract

We demonstrate that $n$-dimension closed Einstein manifolds, whose smallest eigenvalue of the curvature operator of the second kind of $\mathring{R}$ satisfies $\lambda_1 \ge -\theta(n) \bar\lambda$, are either flat or round spheres, where $\bar \lambda$ is the average of the eigenvalues of $\mathring{R}$, and $\theta(n)$ is defined as in equation (1.2). Our result improves a celebrated result (Theorem 1.1) concerning Einstein manifolds with nonnegative curvature operator of the second kind.