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)?