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. 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. Let G ⊂ C n {displaystyle Gsubset {mathbb {C} }^{n}} be a domain and let K ( z , w ) {displaystyle K(z,w)} be the Bergman kernelon G. We define a Hermitian metric on the tangent bundle T z C n {displaystyle T_{z}{mathbb {C} }^{n}} by for z ∈ G {displaystyle zin G} . Then the length of a tangent vector ξ ∈ T z C n {displaystyle xi in T_{z}{mathbb {C} }^{n}} isgiven by This metric is called the Bergman metric on G. The length of a (piecewise) C1 curve γ : [ 0 , 1 ] → C n {displaystyle gamma colon o {mathbb {C} }^{n}} isthen computed as The distance d G ( p , q ) {displaystyle d_{G}(p,q)} of two points p , q ∈ G {displaystyle p,qin G} is then defined as 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 underbiholomorphic mappings of G to another domain G ′ {displaystyle G'} . That is if fis a biholomorphism of G and G ′ {displaystyle G'} , then d G ( p , q ) = d G ′ ( f ( p ) , f ( q ) ) {displaystyle d_{G}(p,q)=d_{G'}(f(p),f(q))} . This article incorporates material from Bergman metric on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.