Connections on almost complex Finsler manifolds and Kobayashi hyperbolicity

Abstract

In this paper, we stablish a necesary condition in terms ofcurvature for the Kobayashi hyperbolicity of a clas of almo st complexFinsler manifolds. For an almost complex Finsler manifold with the con-dition (R), so-caled Riza manifold, we show that here xi sts a uniqueconection compatible with the metric and the almost comple x structurewhich as the horizontal torsion in a special form. With this conection,we dene a holomorphic sectional curvature. Then we show that hisholomorphic sectional curvature of an almost complex subma nifold is notgreater than that of the ambient manifold. This fact, in turn, impliesthat a Riza manifold is hyperbolic if its holomorphic secti onal curvatureis bounded above by 1.