Recent results on complex Cartan spaces

In this paper, we first provide an updated survey of the geometry of complex Cartan spaces. New characterizations for some particular classes of complex Cartan spaces are pointed out, e.g. Landsberg–Cartan, strongly Berwald–Cartan and others. We introduce the Cartan–Randers spaces which offer examples of Berwald–Cartan and strongly Berwald–Cartan spaces. Then, we investigate the complex geodesic curves of a complex Cartan space, using the image by Legendre transformation (LL-duality) of complex geodesic curves of a complex Finsler space. Assuming the weakly Kähler condition for a complex Cartan space, we establish that its complex geodesic curves derive from Hamilton–Jacobi equations. Also, by LL-duality, we introduce the corespondent notion of the projectively related complex Finsler metrics, on the complex Cartan spaces. Various descriptions of the projectively related complex Cartan metrics are given. As applications, the projectiveness of a complex Cartan–Randers metric and the locally projectively flat complex Cartan metrics are analyzed.