# Kobayashi distance function on the upper half-space

Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ matrices with positive-definite imaginary part. Royden in his article "inavariant metrics on Teichmüller space" gives the inifinitesimal form of the Kobayashi distance (metric) on $\mathbb{H}_{g}$ as the operator norm $\frac{1}{2}||Y^{-1/2} dZ Y^{-1/2}||$. My question is what is the Kobayashi distance between two given points on $\mathbb{H}_{g}$? For example, when $g=1$, (i.e., $\mathbb{H}_{g}=\mathbb{H}_{1}$ is the upper half plane) it is well-konwn that the Kobayashi distance (finite version of the Kobayashi metric given above) between $z$ and $w$ in $\mathbb{H}$ is the usual hyperbolic distance $d_{\mathbb{H}}(z,w)$. What is the generalization of this to the upper half space?
• Thank you very much. I just did not understand why the metrics discussed there ($d_p$ or $d_{\infty}$) is exactly the finite version of the Kobayashi metric? Is it strightforward to see this?