Rigidity of Einstein Manifolds under Two Eigenvalue Conditions on the Curvature Operator of the Second Kind

We prove a rigidity theorem for closed Einstein manifolds of dimension $n\ge 6$ under two lower bounds on the curvature operator of the second kind. More precisely, for $L_1,L_2\ge 0$ and some real number $\alpha >1$ in a suitable range, we consider $${\lambda }_1\ge -L_1\bar{\lambda },\frac{1}{\alpha }\sum _{j=1}^{\alpha }{\lambda }_j\ge -L_2\bar{\lambda }\mathrm{.}$$ Here ${\lambda }_1\le \cdots \le {\lambda }_N$ are the eigenvalues of the curvature operator of the second kind $\mathring{R}$, $N=(n-1)(n+2)\mathrm{/}2$, and $\bar{\lambda }=N^{-1}{\sum }_{j=1}^N{\lambda }_j$. Let $\theta (n,\alpha )$ be the constant appearing in \cite[Theorem~1.1]{CW26}; see (2). In the parameter range considered here, $L_2>\theta (n,\alpha )$, so the second condition is strictly weaker than the corresponding condition in \cite{CW26}, while the first condition and that condition do not imply each other. Under an additional explicit relation between $L_1$ and $L_2$, we prove that the manifold is either flat or a spherical space form.