h-projective geometry on compact Kaehler manifolds

The basic geometric structure in h-projective geometry is the family of h-planar curves, associated to a given Kaehler metric. Such curves can be seen as generalisations of geodesics on Kaehler manifolds. In this context, one problem of interest is the investigation of Kaehler manifolds admitting the existence of another Kaehler metric having the same h-planar curves as the given one. One result which i want to present in my talk is a classical conjecture attributed to Yano and Obata and was obtained in a joint work with V. S. Matveev: every compact Kaehler manifold which admits the existence of an h-projective vector field (that is a one-parameter group of transformations preserving the set of h-planar curves) is isomorphic to the complex projective space with Fubini-Study metric provided the h-projective vector field is not a Killing vector field. A second result, obtained in a joint work with A. Fedorova, V. Kiosak and V. S. Matveev, deals with a compact Kaehler manifold, such that the dimension of the space of metrics sharing the same h-planar curves as the given metric is bigger than two. Such a manifold is either isomorphic to the complex projective space with Fubini-Study metric or the Levi-Civita connections of the metrics, having the same h-planar curves as the given metric, coincide.