differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, the Bergman metric is a

Hermitian metric In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on ea ...

that can be defined on certain types of

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

. It is so called because it is derived from the

Bergman kernel In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is the reproducing kernel for the Hilbert space (RKHS) of all square integrable holomorphic functions on a domain ''D'' in C''n''. In deta ...

, both of which are named after

Stefan Bergman Stefan Bergman (5 May 1895 – 6 June 1977) was a Congress Poland-born American mathematician whose primary work was in complex analysis. His name is also written Bergmann; he dropped the second "n" when he came to the U. S. He is best known for t ...

Definition

Let

G \subset ^n

be a domain and let

K(z,w)

be the

on ''G''. We define a Hermitian metric on the

tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...

T_z^n

by :

g_ (z)
:=
\frac
\log K(z,z) ,

for

z \in G

. Then the length of a tangent vector

\xi \in T_z^n

is given by :

\left\vert \xi \right\vert_:=\sqrt.

This metric is called the Bergman metric on ''G''. The length of a (piecewise) ''C''¹ curve

\to ^n

is then computed as :

\ell (\gamma) =
\int_0^1 \left\vert \frac(t) \right\vert_ dt .

The distance

d_G(p,q)

of two points

p,q \in G

is then defined as :

d_G(p,q):=
\inf \ .

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 In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic. Formal definiti ...

mappings of ''G'' to another domain

G'

. That is if ''f'' is a biholomorphism of ''G'' and

G'

, then

d_G(p,q) = d_(f(p),f(q))

References

* Steven G. Krantz. ''Function Theory of Several Complex Variables,'' AMS Chelsea Publishing, Providence, Rhode Island, 1992. Complex manifolds {{differential-geometry-stub