Invariant distances related to the Bergman function

Abstract:

Let $D$ be a bounded domain in ${{\mathbf {C}}^n}$. The invariant distance in $D$ is given by \[ {\rho _D}(z,w) = {\left ( {1 - {{\left ( {\frac {{{K_D}(z,w){K_D}(w,z)}}{{{K_D}(z,z){K_D}(w,w)}}} \right )}^{1/2}}} \right )^{1/2}}.\] It is shown that one half of the length of a piecewise ${C^1}$ curve $\gamma :[a,b] \to D$ with respect to the Bergman metric is equal to the length of $\gamma$ measured by ${\rho _D}$, which implies that the associated inner distance $\rho _D^*$ coincides (up to the factor $\tfrac {1}{2}$) with the Bergman-distance. Also it was proved that ${\rho _D}$ is not an inner distance.