Generalizations of Tachibana’s theorem

Abstract:

A famous theorem of Tachibana says that compact Einstein manifolds with positive curvature operators have constant curvature. In this talk we will discuss several generalizations of this theorem. For example, we show that it suffices to assume that the curvature operator is $\lfloor \frac{n-1}{2} \rfloor$-positive, where $n$ is the dimension of the manifold. Time permitting, we discuss analogues of Tachibana’s theorem for K"ahler manifolds and quaternion K"ahler manifolds. This talk is based on joint work with Peter Petersen.