New rigidity theorem of Einstein manifolds and curvature operator of the second kind

Using Bochner techniques, we prove that a compact Einstein manifold of dimension $n \ge 4$ has constant curvature provided that the curvature operator of the second kind satisfies a cone condition that is strictly weaker than nonnegativity. Furthermore, employing a result of Li \cite{Li5}, we establish that any closed Einstein manifold of dimension $n \ge 4$ satisfying $$k^{-1}({\lambda }_1+\cdots +{\lambda }_k)\ge -\theta(n,k) \bar{\lambda },\quad \text{for some} \quad k \le [\frac{n+2}{4}]$$ must be either flat or a spherical space form. Here, ${\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_{\frac{(n-1)(n+2)}{2}}$ are the eigenvalues of $\mathring{R}\,$, $\bar{\lambda }$ is their average, and $\theta (n,k)$ is a positive constant. This result generalizes the work of Dai-Fu \cite{DF} and Chen-Wang \cite{CW1,CW}.We also classify four-dimensional Einstein manifolds satisfying a cone condition.