Deformation of complex Finsler metrics

The aim of this paper is to describe the infinitesimal deformation (M, V) of a complex Finsler space family {(M, Lt)}t∈ℝ and to study some of its geometrical objects (metric tensor, non-linear connection, etc). In this circumstances the induced non-linear connection on (M, V) is defined. Moreover we have elaborate the inverse problem, the problem of the first order deformation of the metric. A special part is devoted to the study of particular cases of the perturbed metric.

On complex Berwald metrics which are not conformal changes of complex Minkowski metrics

We investigate a class of complex Finsler metrics on a domain D ⊂ ℂn. Necessary and sufficient conditions for these metrics to be strongly pseudoconvex complex Finsler metrics, or complex Berwald metrics, are given. The complex Berwald metrics constructed in this paper are neither trivial Hermitian metrics nor conformal changes of complex Minkowski metrics. We give a characterization of complex Berwald metrics which are of isotropic holomorphic curvatures, and also give characterizations of complex Finsler metrics of this class to be Kähler Finsler or weakly Kähler Finsler metrics. Moreover, in the strongly convex case, we give characterizations of complex Finsler metrics of this class to be projectively flat Finsler metrics or dually flat Finsler metrics.

NEW CANDIDATES FOR A HERMITIAN APPROACH OF GRAVITY

In this paper, some possible candidates for the study of gravity are proposed in terms of complex Finsler geometry. These mainly concern the complex Hermitian versions of weakly gravitational metric and Schwarzschild metric. For the weakly gravitational fields, we state few interesting geometrical and physical aspects such as the conditions under which a complex Finsler metrics are projectively related to the weakly gravitational metric. In the Kähler case, the geodesic curves of the weakly gravitational metric are obtained. Some applications concerning the deformations of the weakly gravitational Hermitian metric to a complex Randers metric are described. Another candidate for gravity is given by so-called Hermitian Schwarzschild metric for which some geodesic curves are highlighted. The last part of the paper is devoted to a generalization of the complex Klein–Gordon equations, in terms of Quantum field theory on a curved space.