The Tanno-Theorem for K\"ahlerian metrics with arbitrary signature

Abstract

Considering a non-constant smooth solution $f$ of the Tanno equation on a closed, connected K\"ahler manifold $(M,g,J)$ with positively definite metric $g$, Tanno showed that the manifold can be finitely covered by $(\mathbb{C}P(n),\mbox{const}\cdot g_{FS})$, where $g_{FS}$ denotes the Fubini-Study metric of constant holomorphic sectional curvature equal to $1$. The goal of this paper is to give a proof of Tannos Theorem for K\"ahler metrics with arbitrary signature.