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

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.

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References in corpus (4)

• Betti numbers and the curvature operator of the second kind
• Manifolds with nonnegative curvature operator of the second kind
• Holonomy restrictions from the curvature operator of the second kind
• Product manifolds and the curvature operator of the second kind