Recent results on complex Cartan spaces

TL;DR

This paper surveys the geometry of complex Cartan spaces, introduces new classes and examples, and explores their geodesic curves, dualities, and projective properties, providing new insights and characterizations in complex differential geometry.

Contribution

It offers updated classifications, introduces Cartan-Randers spaces, and examines geodesic and projective properties of complex Cartan spaces using Legendre duality.

Findings

Complex geodesic curves relate to Hamilton-Jacobi equations under Kähler conditions.

Cartan-Randers spaces serve as examples of Berwald-Cartan spaces.

Projectively related complex Cartan metrics are characterized and analyzed.

Abstract

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 ($\mathcal{L}-$ duality) of complex geodesic curves of a complex Finsler space. Assuming the weakly K\"{a}hler condition for a complex Cartan space, we establish that its complex geodesic curves derive from Hamilton-Jacobi equations. Also, by $\mathcal{L}-$ 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.