In differential geometry, the Bergman metric is a Hermitian metric that can be defined on certain types of complex manifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

. It is so called because it is derived from the Bergman kernel

Bergman kernel

In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is a reproducing kernel for the Hilbert space of all square integrable holomorphic functions on a domain D in Cn....

, both of which are named for Stefan Bergman

Stefan Bergman

Stefan Bergman was a Polish-born American mathematician whose primary work was in complex analysis. He is best known for the kernel function he discovered while at Berlin University in 1922. This function is known today as the Bergman kernel...

.

Definition

Let be a domain and let be the Bergman kernel
on G. We define a Hermitian metric on the tangent bundle

Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...

  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 (piecewise) C¹ curve

Smooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

  is
then computed as


The distance of two points is then defined as


The distance d_G 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 d_G is invariant under
biholomorphic mappings of G to another domain . That is if f
is a biholomorphism of G and , then .