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 G \subset {\mathbb{C}}^n be a domain and let K(z,w) be the Bergman kernel on G. We define a Hermitian metric on the tangent bundle T_z{\mathbb{C}}^n by


g_{ij} (z)
:=
\frac{\partial^2}{\partial z_i\, \partial \bar{z}_j}
\log K(z,z) ,

for z \in G. Then the length of a tangent vector \xi \in T_z{\mathbb{C}}^n is given by

\left\vert \xi \right\vert_{B,z}:=\sqrt{\sum_{i,j=1}^n g_{ij}(z) \xi_i \bar{\xi}_j }.

This metric is called the Bergman metric on G.

The length of a (piecewise) C1 curve \gamma \colon [0,1] \to {\mathbb{C}}^n is then computed as


\ell (\gamma) =
\int_0^1 \left\vert \frac{\partial \gamma}{\partial t}(t) \right\vert_{B,\gamma(t)} dt .

The distance d_G(p,q) of two points p,q \in G is then defined as


d_G(p,q):=
\inf \{ \ell (\gamma) \mid \text{ all piecewise }C^1\text{ curves }\gamma\text{ such that }\gamma(0)=p\text{ and }\gamma(1)=q \} .

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 G'. That is if f is a biholomorphism of G and G', then d_G(p,q) = d_{G'}(f(p),f(q)).

References

This article incorporates material from Bergman metric on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This article is issued from Wikipedia - version of the Sunday, March 02, 2014. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.