Bergman metric


In differential geometry, the Bergman metric is a Hermitian metric that can be defined on certain types of complex manifold. It is so called because it is derived from the Bergman kernel, both of which are named for Stefan Bergman.

Definition

Let be a domain and let be the Bergman kernel
on G. We define a Hermitian metric on the tangent bundle by
for. Then the length of a tangent vector is
given by
This metric is called the Bergman metric on G.
The length of a C1 curve is
then computed as
The distance of two points is then defined as
The distance dG is called the Bergman distance.
The Bergman metric is in fact a positive definite matrix at each point if G is a bounded domain. More importantly, the distance dG is invariant under
biholomorphic mappings of G to another domain. That is if f
is a biholomorphism of G and, then.