J. Korean Math. Soc. 2015; 52(3): 587-601
Printed May 1, 2015
https://doi.org/10.4134/JKMS.2015.52.3.587
Copyright © The Korean Mathematical Society.
Nicoleta Aldea and Gabriela C\^{a}mpean
Transilvania University, Transilvania University
In this paper, we investigate the $\mathbb{R}$-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 $\mathbb{R}$-complex Hermitian Finsler spaces are defined, (e.g. weakly K\"{a}hler, K\"{a}hler, strongly K\"{a}hler). Here the notions of K\"{a}hler and strongly K\"{a}hler 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 $\mathbb{R}$-complex Hermitian Finsler space to become a weakly Berwald or Berwald. Finally, we obtain the conditions under which an $\mathbb{R}$-complex Hermitian Finsler space with Randers metric is Berwald. We get some clear examples which illustrate the interest for this work.
Keywords: $\mathbb{R}$-complex Hermitian Finsler space, Berwald space, Randers space
MSC numbers: 53B40, 53C60
2000; 37(1): 73-84
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