Let M be a complex manifold
of dimension n furnished with both the Bergman metric and the Carathéodory
distance. The main result of the present paper is to prove that the Bergman
metric is always greater than or equal to the Carathéodory distance on
M. The case where M is a bounded domain in the space Cn was already
considered by the author in Proc. Nat. Acad. Sci. (U.S.A.), 73 (1976),
4294.