Recently I just learned the Kobayashi distance on complex manifolds and wants to get some feeling of how it looks like on exmaples of manifolds with positive Ricci curvature. I have a feeling that the Kobayashi distance on those manifolds should vanish since those are not very "hyperbolic".


A simple example is extended complex plane. then its Kobayashi distance vanishes since the automorphism group can contract one point very close to the origin while keeping the origin fixed. I believe it is similarly true for all complex projective spaces with usual complex structure since automorphism group is known. Can we have more examples? Or is there a counterexample (an example of Kaehler manifold with positive Ricci curvature but not identically vanishing Kobayashi distance)?



How to compute Kobayashi distance of compact Kaehler manifolds with postive Ricci curvature? 
Positive Ricci curvature Kaehler manifolds are Fano, and therefore rationally connected. See Janos Kollar's book [Rational Curves on Algebraic Varieties](http://books.google.com/books?hl=en&lr=&id=oqW3GabJLjgC&oi=fnd&pg=PA1&dq=janos+kollar+rational+curves&ots=_a9ah5mhdV&sig=-xKWuKioqlb6J_U_L_bxP8Kd-Wg#v=onepage&q=janos%2520kollar%2520rational%2520curves&f=false). Therefore any two points lie on a rational curve. Since the Kobayashi pseudodistance along a subvariety is never less than the pseudodistance in the ambient space, any pair of points lie at 0 pseudodistance.


