  The purpose of this article is to consider two themes both of which emanate
from and involve the Kobayashi and the Carath\'eodory metric. First we study
the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex
domains, on weakly pseudoconvex domains of finite type in $\mathbf C^2$ and on
convex finite type domains in $\mathbf C^n$ using the scaling method.
Applications include an alternate proof of the Wong-Rosay theorem, a
characterization of analytic polyhedra with noncompact automorphism group when
the orbit accumulates at a singular boundary point and a description of the
Kobayashi balls on weakly pseudoconvex domains of finite type in $
\mathbf{C}^2$ and convex finite type domains in $ \mathbf{C}^n $ in terms of
Euclidean parameters. Second a version of Vitushkin's theorem about the uniform
extendability of a compact subgroup of automorphisms of a real analytic
strongly pseudoconvex domain is proved for $C^1$-isometries of the Kobayashi
and Carath\'eodory metrics on a smoothly bounded strongly pseudoconvex domain.
