Stefan Rosemann

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

Abstract Two Kahler metrics on a complex manifold are called c-projectively equivalent if their J-planar curves coincide. Such curves are defined by the property that the acceleration is complex proportional to the velocity. The degree of mobility of a Kahler 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 a simply connected Kahler manifold by reducing the problem to the study of parallel Hermitian ( 0 , 2 ) -tensors on the conification of the manifold. We also describe all such values for a Kahler–Einstein metric. We apply these results to describe all possible dimensions of the space of essential c-projective vector fields of Kahler and Kahler–Einstein metrics. We also show that two c-projectively equivalent Kahler–Einstein metrics (of arbitrary signature) on a closed manifold have constant holomorphic curvature or are affinely equivalent.

Curvature and the c-projective mobility of Kaehler metrics with hamiltonian 2-forms

The mobility of a Kaehler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kaehler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kaehler metric to have mobility at least three: its curvature must have nontrivial nullity at every point. Using the local classification of Kaehler metrics with hamiltonian 2-forms, we describe explicitly the Kaehler metrics with mobility at least three and hence show that the nullity condition on the curvature is also sufficient, up to some degenerate exceptions. In an Appendix, we explain how the classification may be related, generically, to the holonomy of a complex cone metric.

The degree of mobility of Einstein metrics

Abstract Two (pseudo-)Riemannian metrics are called projectively equivalent if their unparametrized geodesics coincide. The degree of mobility of a metric is the dimension of the space of metrics that are projectively equivalent to it. We give a complete list of possible values for the degree of mobility of Riemannian and Lorentzian Einstein metrics on simply connected manifolds, and describe all possible dimensions of the space of essential projective vector fields.