Einstein manifolds and curvature operator of the second kind

Abstract

We prove that a compact Einstein manifold of dimension $n\ge 4$ with nonnegative curvature operator of the second kind is a constant curvature space by Bochner technique. Moreover, we obtain that compact Einstein manifolds of dimension $n\ge 11$ with $\left[ \frac{n+2}{4} \right]$-nonnegative curvature operator of the second kind, $4\ (\text{ resp. },8,9,10)$-dimensional compact Einstein manifolds with 2-nonnegative curvature of the second kind and 5-dimensional compact Einstein manifolds with 3-nonnegative curvature of the second kind are constant curvature spaces. Combining with Li’s (J Geom Anal 32:281, 2022) result, we have that a compact Einstein manifold of dimension $n\ge 4$ with $\max \{4,\left[ \frac{n+2}{4} \right] \}$-nonnegative curvature operator of the second kind is a constant curvature space.