• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.237-247
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• 2007

In this paper, we establish a necessary condition in terms of curvature for the Kobayashi hyperbolicity of a class of almost complex Finsler manifolds. For an almost complex Finsler manifold with the condition (R), so-called Rizza manifold, we show that there exists a unique connection compatible with the metric and the almost complex structure which has the horizontal torsion in a special form. With this connection, we define a holomorphic sectional curvature. Then we show that this holomorphic sectional curvature of an almost complex submanifold is not greater than that of the ambient manifold. This fact, in turn, implies that a Rizza manifold is hyperbolic if its holomorphic sectional curvature is bounded above by -1.

• The Pure and Applied Mathematics
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• v.5 no.1
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• pp.73-78
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• 1998

Let $H_1$ ($\Delta$, M) be the family of all 1-1 holomorphic mappings of the unit disk $\Delta\; \subset\; C$ into a complex manifold M. Following the method of Royden, Hahn introduces a new pseudo-differential metric $S_{M}$ on M. The present paper is to study the product property of the metric $S_{M}$ when M is given by the product of two domains $D_1$ and $D_2$ in the complex plane C, thus investigating the hyperbolicity of the product domain $D_1 \;\times\; D_2$ with respect to $S_{M}$ metric.