Concept
In mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyperbolic manifolds are an important class of complex manifolds, defined by the property that the Kobayashi pseudometric is a metric. Kobayashi hyperbolicity of a complex manifold X implies that every holomorphic map from the complex line C to X is constant. The origins of the concept lie in Schwarz's lemma in complex analysis. Namely, if f is a holomorphic function on the open unit disc D in the complex numbers C such that f(0) = 0 and |f(z)| < 1 for all z in D, then the derivative f '(0) has absolute value at most 1. More generally, for any holomorphic map f from D to itself (not necessarily sending 0 to 0), there is a more complicated upper bound for the derivative of f at any point of D. However, the bound has a simple formulation in terms of the Poincaré metric, which is a complete Riemannian metric on D with curvature −1 (isometric to the hyperbolic plane). Namely: every holomorphic map from D to itself is distance-decreasing with respect to the Poincaré metric on D. This is the beginning of a strong connection between complex analysis and the geometry of negative curvature. For any complex space X (for example a complex manifold), the Kobayashi pseudometric dX is defined as the largest pseudometric on X such that for all holomorphic maps f from the unit disc D to X, where denotes distance in the Poincaré metric on D. In a sense, this formula generalizes Schwarz's lemma to all complex spaces; but it may be vacuous in the sense that the Kobayashi pseudometric dX may be identically zero. For example, it is identically zero when X is the complex line C. (This occurs because C contains arbitrarily big discs, the images of the holomorphic maps fa: D → C given by f(z) = az for arbitrarily big positive numbers a.) A complex space X is said to be Kobayashi hyperbolic if the Kobayashi pseudometric dX is a metric, meaning that dX(x,y) > 0 for all x ≠ y in X.