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Two Kaehler metrics on one complex manifold are said to be c-projectivelyequivalent if their J-planar curves, i.e., curves defined by the property thattheir acceleration is complex proportional to their velocity, coincide. Thedegree of mobility of a Kaehler metric is the dimension of the space of metricsthat are c-projectively equivalent to it. We give the list of all possiblevalues of the degree of mobility of simply connected 2n-dimensional RiemannianKaehler manifolds. We also describe all such values under the additionalassumption that the metric is Einstein. As an application, we describe allpossible dimensions of the space of essential c-projective vector fields ofKaehler and Kaehler-Einstein Riemannian metrics. We also show that twoc-projectively equivalent Kaehler Einstein metrics (of arbitrary signature) ona closed manifold have constant holomorphic curvature or are affinelyequivalent.

Conification construction for Kaehler manifolds and its application in c-projective geometry