The only Kähler manifold with degree of mobility at least 3 is \((\mathbb CP(n), g_{\text{Fubini-Study}})\) (Q2901908)

Let \(g\) be a (possible non positive definite) Kähler metric on a complex manifold \(M\) with complex structure \(J.\) A curve \(\gamma:I\rightarrow M\) is called \(h\)-planar is there are functions \(\alpha(t),\beta(t)\) such that NEWLINE\[NEWLINE\nabla _{\gamma'(t)}{\gamma'(t)}=\alpha(t)\gamma'(t)+\beta(t)J(\gamma'(t)).NEWLINE\]NEWLINE Two Kähler metrices \(g\) and \(\bar{g}\) with the same complex structure are called \(h\)-projectively equivalent, if each \(h\)-planar curve of \(g\) is an \(h\)-planar curve of \(\bar{g}\) and vice versa. The degree of mobility of a Kähler metric \(g\) measures the dimension of the space of Kähler metrics which are \(h\)-projectively equivalent to \(g\).NEWLINENEWLINEThe paper under review proves that a metric on a closed connected manifold can not 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 main application it proves 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 (\({\mathbb C}{\mathbb P}(n), g_{\text{Fubini.Study}})\) (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.