Shoshichi Kobayashi

Intrinsic metrics on complex manifolds

1. Definition of intrinsic pseudometric. Let M be a (connected) complex manifold. We shall define a pseudometric d on M in a natural manner so that it depends only on the complex structure of M and nothing else. Let D be the open unit disk in the complex plane and p the distance on D defined by the Poincaré-Bergman metric of D. Given two points p and q of M, choose the following objects: (1) points p=p0t pu • • • , pk-i, pk=q of M and (2) points ai, • • • , a^ bi, • • • , bu of D and holomorphic mappings fu • • * » fk of D into M such that ƒ»•(#»•)—^-î and fiipd—pi for i = l, • • • , &. For each choice of points and mappings satisfying (1) and (2), consider the number p(#i, &i)+ • • • +p(flto, &&)• Let d(£, #) be the infimum of the numbers obtained in this manner for all possible choices. I t is easy to verify that d is a pseudometric on M in the sense that

A theorem on filtered Lie algebras and its applications

The purpose of this note is to announce the following theorem and indicate a few of its applications to the classification of irreducible (complex or real) Lie algebras of linear endomorphisms of finite type of order 2 (or higher) as well as some questions on transformation groups acting on symmetric spaces. Other applications to homogeneous spaces as well as the proofs of all the results stated below will be published elsewhere.