Finsler Geometry on Complex Vector Bundles

A Finsler metric of a manifold or vector bundle is defined as a smooth as- signment for each base point a norm on each fibre space, and thus the class of Finsler metrics contains Riemannian metrics as a special sub-class. For this reason, Finsler geometry is usually treated as a generalization of Riemannian ge- ometry. In fact, there are many contributions to Finsler geometry which contain Riemannian geometry as a special case (see e.g., [Bao et al. 2000], [Matsumoto 1986], and references therein). On the other hand, we can treat Finsler geometry as a special case of Riemann- ian geometry in the sense that Finsler geometry may be developed as differential geometry of fibred manifolds (e.g., [Aikou 2002]). In fact, if a Finsler metric in the usual sense is given on a vector bundle, then it induces a Riemannian inner product on the vertical subbundle of the total space, and thus, Finsler geometry is translated to the geometry of this Riemannian vector bundle. It is natural to question why we need Finsler geometry at all. To answer this question, we shall describe a few applications of complex Finsler geometry to some subjects which are impossible to study via Hermitian geometry. The notion of complex Finsler metric is old and goes back at least to Carathéodory who introduced the so-called Carathéodory metric . The geometry of complex Finsler manifold, via tensor analysis, was started by [Rizza 1963], and afterwards, the connection theory on complex Finsler manifolds has been developed by [Rund 1972], [Icijyō 1994], [Fukui 1989], and [Cao and Wong 2003], etc..