ON THE DEFINITION OF THE KOBAYASHI–BUSEMAN PSEUDOMETRIC

We prove that the (2n - 1)th Kobayashi pseudometric of any domain D ⊂ ℂn coincides with the Kobayashi–Buseman pseudometric of D, and that 2n - 1 is the optimal number, in general.

On the Kobayashi Pseudometric Reduction of Homogeneous Spaces

Given any homogeneous complex manifold X = G/H, there exists a natural coset map π :G/H → G/K satisfying π (X1) = π (x2) if and only if dx(x1 x2) = 0, where dx denotes the Kobayashi pseudometric on X. Its typical fiber Z : = K/H is a connected complex submanifold of X. Also G/K has a (7-invariant complex structure, provided K satisfies a certain technical assumption (see Theorem 3). If Z is compact as well, then G/K is biholomorphic to a homogeneous bounded domain.