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.

我们通过Bochner技术证明了具有第​​二类非负曲率算子的维数\(n\ge 4\)的紧爱因斯坦流形是一个常曲率空间。此外，我们得到维数为\(n\ge 11\)的紧爱因斯坦流形，其中\(\left[ \frac{n+2}{4} \right] \) -第二类非负曲率算子，\(具有第二类 2-非负曲率的 4\ (\text{ resp. },8,9,10)\)维紧爱因斯坦流形和具有第二类 3-非负曲率的 5 维紧爱因斯坦流形是常数曲率空间。结合 Li 的 (J Geom Anal 32:281, 2022) 结果，我们有一个维数为\(n\ge 4\)的紧爱因斯坦流形，其中\(\max \{4,\left[ \frac{n+2 }{4} \right] \}\) -第二类非负曲率算子是常曲率空间。