## The Chern-Finsler connection and Finsler-Kähler manifolds.(English)Zbl 1144.53030

Sabau, Sorin V. (ed.) et al., Finsler geometry, Sapporo 2005. In memory of Makoto Matsumoto. Proceedings of the 40th Finsler symposium on Finsler geometry, Sapporo, Japan, September 6–10, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-42-6/hbk). Advanced Studies in Pure Mathematics 48, 343-373 (2007).
In the framework of complex Finsler geometry, the author studies the Chern-Finsler connection $$\nabla$$ and outlines its applications. The main addressed issues are: (i) the ampleness of holomorphic vector bundles over compact complex manifolds – based on the study [S. Kobayashi, Nagoya Math. J. 57, 153–166 (1975; Zbl 0326.32016)], (ii) the characterization of a certain class of complex Finsler metrics in terms the torsion and curvature of $$\nabla$$, and (iii) the characterization of Finsler-Kähler manifolds in terms of the Cartan connection naturally induced from $$\nabla$$ on the real tangent space of the manifold.
For the entire collection see [Zbl 1130.53005].

### MSC:

 53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics) 53C56 Other complex differential geometry 53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)

Zbl 0326.32016