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

TL;DR

This paper classifies the possible degrees of mobility for simply connected Kähler manifolds and explores their c-projective equivalence, especially in Einstein metrics, with applications to c-projective vector fields and curvature properties.

Contribution

It provides a complete classification of the degree of mobility for Kähler manifolds and describes the structure of c-projectively equivalent Einstein metrics.

Findings

Classified all possible degrees of mobility for simply connected 2n-dimensional Kähler manifolds.

Described the degrees of mobility under the Einstein condition.

Showed that c-projectively equivalent Einstein metrics on closed manifolds have constant holomorphic curvature or are affinely equivalent.

Abstract

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