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By Pindar.; Theodor Bergk

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So then, which condition will ensure positive-definiteness of the Kobayashi distance? L. Royden in [Royden 1971]. 1 of these lecture notes), the infinitesimal Kobayashi metric (or, as it is often called the Kobayashi metric (or, the Kobayashi-Royden metric)) kM : T M → R of M is defined to be kM (p; v) = inf{|λ| : ∃h ∈ Hol (D, M ) such that h(0) = p, dh0 (λ) = v}. 1 The research concerning Kobayashi distance and metric became so extensive over decades; see [Kobayashi 1998] and references therein.

Of course it is worth reading the original text ([Chern 1968], [Lu 1968]).

Greene and Wu 1979]). Also, the Hermitian metric on M need not be complete in order for the proposition to be valid. September 9, 2010 18:12 World Scientific Book - 9in x 6in Ahlfors’ Generalization schwarzs 21 Following [Kobayashi 1967a], we call a complex manifold hyperbolic (or more precisely, hyperbolic in the sense of Kobayashi), if its Kobayashi distance is positive-definite. Call a complex manifold complete hyperbolic if its Kobayashi distance is complete in the sense that all Cauchy sequences converge.

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