  In his famous 1981 paper, Lempert proved that given a point in a strongly
convex domain the complex geodesics (i.e., the extremal disks) for the
Kobayashi metric passing through that point provide a very useful fibration of
the domain. In this paper we address the question whether, given a smooth
complex Finsler metric on a complex manifold, it is possible to give purely
differential geometric properties of the metric ensuring the existence of such
a fibration in complex geodesics of the manifold. We first discuss at some
length the notion of holomorphic sectional curvature for a complex Finsler
metric; then, using the differential equation of complex geodesics we obtained
in a previous paper, we show that for every pair (point, tangent vector) there
is a (only a segment if the metric is not complete) complex geodesic passing
through the point tangent to the given vector iff the Finsler metric is
K\"ahler, has constant holomorphic sectional curvature -4 and satisfies a
simmetry condition on the curvature tensor. Finally, we show that a complex
Finsler metric of constant holomorphic sectional curvature -4 satisfying the
given simmetry condition on the curvature is necessarily the Kobayashi metric.
