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

Abstract

In this note, we study Einstein manifolds whose curvature operator of the second kind $\mathring{R}$ satisfies the cone condition \[ α^{-1}\big(\sum_{i=1}^{[α]} λ_i+ (α- [α] ) λ_{[α] + 1} \big) \ge -θ\barλ \] for some real number $\alpha \in [1,(n+2)(n-1)/2)$. Here $[\alpha ]:=\max \{m\in \mathbb{Z}:m\le \alpha \}$, $\theta >-1$ 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}.

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