The Hermitian Connection and the Jacobi Fields of a Complex Finsler   Manifold
  It is proved that all invariant functions of a complex Finsler manifold can be totally recovered from the torsion and curvature of the connection introduced by Kobayashi for holomorphic vector bundles with complex Finsler structures. Equations of the geodesics and Jacobi fields of a generic complex Finsler manifold, expressed by means Kobayashi's connection, are also derived. 
A. Spiro