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.

Definition

Let $G\; subset\; \{mathbb\{C^n$ be a domain and let $K(z,w)$ be the Bergman kernelon "G". We define a Hermitian metric on the tangent bundle $T\_z\; \{mathbb\{C^n$ by:for $z\; in\; G$. Then the length of a tangent vector $xi\; in\; T\_z\{mathbb\{C^n$ isgiven by

:

This metric is called the Bergman metric on "G".

The length of a (piecewise) "C"¹ curve $gamma\; colon\; [0,1]\; o\; \{mathbb\{C^n$ isthen computed as

:$ell\; (gamma)\; =int\_0^1\; leftvert\; frac\{partial\; gamma\}\{partial\; t\}(t)\; ightvert\_\{B,gamma(t)\}\; dt\; .$

The distance $d\_G(p,q)$ of two points $p,q\; in\; G$ is then defined as

:$d\_G(p,q):=inf\; \{\; ell\; (gamma)\; mid\; ext\{\; all\; piecewise\; \}C^1\; ext\{\; curves\; \}gamma\; ext\{\; such\; that\; \}gamma(0)=p\; ext\{\; and\; \}gamma(1)=q\; \}\; .$

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 $G\text{'}$. That is if "f"is a biholomorphism of "G" and $G\text{'}$, then $d\_G(p,q)\; =\; d\_\{G\text{'}\}(f(p),f(q))$.

References

* Steven G. Krantz. "Function Theory of Several Complex Variables," AMS Chelsea Publishing, Providence, Rhode Island, 1992.