On the volume of the projectivized tangent bundle in a complex Finsler manifold

Abstract

We prove that the volume of PT ^{1, 0} _z M, calculated with respect to a Kähler metric induced by a complex Finsler structure, is a constant. This contrasts sharply with the situation in real Finsler geometry, where the volume of unit tangent sphere at each point x in a real Finsler manifold is in general a function of x. Furthermore, we point out that different metrics have different constants in general.