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==