Complex Finsler Geometry Via the Equivalence Problem on the Tangent Bundle

Abstract

The infinitesimal Kobayashi metric is a real-valued function F _M on the tangent bundle of a complex manifold M. For p ∊ M and v ∊ T _P M,

$${F_m}(p,\upsilon ) = \inf \left\{ {{1 \over r}:\exists {\rm{ holomorphic }}f:{\Delta _r} \to M{\rm{ with }}f(0) = p,f'(0) = \upsilon } \right\}$$

where Δ _r = {z ∊ ℂ : |z| < r}. In general, this is not a Finsler metric. For example, it vanishes identically for M = ℂⁿ. However, it does define a metric in a number of interesting cases, and Lempert showed [12] that it is a smooth Finsler metric with strongly convex indicatrix if M is a smoothly bounded strongly convex domain in ℂⁿ. Using some further geometric characterizations of such Kobayashi metrics given by Lempert, the author was able to derive the following differential geometric characterization of the Kobayashi metric in [11].