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

TL;DR

This paper proves that closed Einstein manifolds satisfying a specific cone condition on their curvature operator of the second kind are either flat or spherical, extending previous results to a broader class of conditions.

Contribution

The authors generalize existing results by establishing that Einstein manifolds meeting a new cone condition on the curvature operator are necessarily flat or round spheres.

Findings

Manifolds satisfying the cone condition are either flat or spherical.

The results extend previous work to a wider class of cone conditions.

The main theorem applies to closed Einstein manifolds of dimension at least 4.

Abstract

In this note, we study Einstein manifolds whose curvature operator of the second kind $\mathring{R}$ satisfies the cone condition \[ \alpha^{-1}\big(\sum_{i=1}^{[\alpha]} \lambda_i+ (\alpha - [\alpha] ) \lambda_{[\alpha] + 1} \big) \ge -\theta \bar{\lambda} \] for some real number $\alpha \in [1, (n+2)(n-1)/2)$. Here $[\alpha] :=\max\{ m \in \mathbb{Z}: m \leq \alpha\}$, $\theta>-1$ and $\lambda_1 \le \cdots \le \lambda_{(n+2)(n-1)/2}$ are the eigenvalues of $\mathring{R}$ and $\bar{\lambda}$ 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 $\alpha \in \mathbb Z_+$ of the authors \cite{CW24-1,CW25-2} and Fu-Lu \cite{FL25}.