ℝ-complex Cartan spaces

Abstract

In this paper, we investigate the ℝ-complex Cartan spaces highlighting some classes of ℝ-complex Hermitian Cartan spaces. In this study, we use Chern-Cartan and canonical connections. Some classes of ℝ-complex Hermitian Cartan spaces are introduced(weakly Kahler-Cartan, Kahler-Cartan, strongly Kahler-Cartan). The conditions for a ℝ-complex Hermitian Cartan space to be weakly Berwald-Cartan or Berwald-Cartan are obtained. In the last section, we emphasize the necessary and sufficient conditions under which an ℝ-complex Hermitian Cartan space with a Randers metric is Berwald-Cartan. We come with some explicit examples to illustrate the interest in these classes of spaces.