ON INFINITESIMAL BEHAVIOR OF THE KOBAYASHI DISTANCE

The condition that the Kobayashi distance between two nearby points in a pseudo-convex domain is realized by the Poincare distance on a sing le analytic disk joining the two points is studied. It is shown that t he condition forces the Kobayashi indicatrix to be convex. Examples of pseudo-convex domains on which this condition fails to hold are given . The (infinitesimal) Kobayashi metric is shown to be a directional de rivative of the Kobayashi distance. It is shown that, if the condition holds near any point of a pseudo-convex domain and if the Kobayashi m etric is a complete Finsler metric of class C-2,then the Kobayashi dis tance between any two points in the domain can be realized by the Poin care distance on a single analytic disk joining the two points.