The only Kähler manifold with degree of mobility at least 3 is (ℂP (n ), ɡ Fubini-Study )

Abstract

The degree of mobility of a (pseudo-Riemannian) Kähler metric is the dimension of the space of metrics h-projectively equivalent to it. We prove that a metric on a closed connected manifold cannot have the degree of mobility at least 3 unless it is essentially the Fubini–Study metric, or the h-projective equivalence is actually the affine equivalence. As the main application, we prove an important special case of the classical conjecture attributed to Obata and Yano, stating that a closed manifold admitting an essential group of h-projective transformations is (ℂP(n), ɡFubini–Study) (up to multiplication of the metric by a constant). An additional result is the generalization of a certain result of Tanno 1978 for the pseudo-Riemannian situation.