New rigidity theorem of Einstein manifolds and curvature operator of the second kind
Haiping Fu, Yao Lu
TL;DR
The paper investigates rigidity of compact Einstein manifolds under cone conditions on the curvature operator of the second kind, proving that weaker-than-nonnegative cones yield constant curvature in many dimensions. By combining Bochner formulas with algebraic minimization arguments and leveraging Li's cone-condition framework, it establishes existence of dimension- and k-dependent constants $\theta(n,k)$ ensuring flatness or spherical space-form geometry for $n\ge4$ and $k\le\left[\frac{n+2}{4}\right]$, with explicit refinements in dimensions $n=4,5$. In four dimensions, a sharp cone condition $\lambda_1+\lambda_2+\lambda_3\ge -3\bar{\lambda}$ leads to a precise list of Einstein 4-manifolds up to scaling, including quotients of $\mathbb{S}^4$, $\mathbb{CP}^2$ with the Fubini–Study metric, and quotients of $\mathbb{S}^2\times\mathbb{S}^2$. The paper also provides a corrected Bochner formula for the curvature operator, enabling a complete 4D classification (Theorem D and Corollary 4.1) and bridging local curvature-operator cone conditions with global geometric and topological rigidity.
Abstract
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 \[k^{-1}({λ}_1+\cdots +{λ}_k)\ge -θ(n,k) \bar{λ},\quad \text{for some} \quad k \le [\frac{n+2}{4}]\] must be either flat or a spherical space form. Here, ${λ}_1\le {λ}_2\le \cdots \le {λ}_{\frac{(n-1)(n+2)}{2}}$ are the eigenvalues of $\mathring{R}\,$, $\bar{λ}$ is their average, and $θ(n,k)$ 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.