ON SOME CLASSES OF R-COMPLEX HERMITIAN FINSLER SPACES

In this paper, we investigate the H-complex Hermitian Finsler spaces, emphasizing the differences that separate them from the complex Finsler spaces. The tools used in this study are the Chern-Finsler and Berwald connections. By means of these connections, some classes of the R-complex Hermitian Finsler spaces are defined, (e.g. weakly Kahler, Kahler, strongly Kahler). Here the notions of Kahler and strongly Kahler do not coincide, unlike the complex Finsler case. Also, some kinds of Berwald notions for such spaces are introduced. A special approach is devoted to obtain the equivalence conditions for an H-complex Hermitian Finsler space to become a weakly Berwald or Berwald. Finally, we obtain the conditions under which an R-complex Hermitian Finsler space with Randers metric is Berwald. We get some clear examples which illustrate the interest for this work.