# Kähler Geodesics

Consider the Kähler manifold in coordinates $(a,b)$ given by the complex Riemannian metric $$\begin{pmatrix} \frac{1}{1-|a|^2}&\frac{1}{1-a\bar{b}}\\\frac{1}{1-\bar{a}b}&\frac{1}{1-|b|^2}\end{pmatrix}.$$

For statistical reasons we require that $(a,b)$ be in the unit disk in the complex plane.

What are the geodesics of this manifold?

• From your notation it seems that $a,b$ are complex variables? So when you say $(a,b)$ be in the unit disk do you mean $a,b\in D\subset \mathbb{C}$ or $(a,b) \in$some subset$\subset \mathbb{C}^2$? May 7, 2013 at 11:38
• I mean that $a,b \in \mathbb{C}$ and $|a|,|b|<1$. May 7, 2013 at 12:59
• (statistics)? 
– Did
May 3, 2014 at 10:13
• @Did This manifold is the statistical manifold corresponding to linear time invariant systems. Such systems are commonly used in signal processing for linear prediction. May 3, 2014 at 13:32
• @William Just google information geometry, you'll see there is a huge connection between Riemannian geometry and statistics. The goto sources are by Amari. As far as Kahler geometry and it's connection to statistics, this is my personal research. I have not published it to date. Apr 29, 2016 at 13:02

$$\Gamma_{\rho\tau}^\mu=\frac{1}{2}g^{\mu\nu}\left(\frac{\partial g_{\nu\rho}}{\partial x^\tau}+\frac{\partial g_{\nu\tau}}{\partial x^\rho}-\frac{\partial g_{\rho\tau}}{\partial x^\nu}\right)$$
where $g_{\mu\nu}$ is the metric tensor you have above. Then for the non-zero symbols you can plug them into the geodesic differential equation:
$$\frac{\partial^2x^\mu}{\partial\sigma^2}+\Gamma_{\rho\tau}^\mu\frac{\partial x^\rho}{\partial\sigma}\frac{\partial x^\tau}{\partial\sigma}=\gamma(\sigma)\frac{\partial x^\mu}{\partial\sigma}$$
where the geodesic is parametrized by $\sigma$ and $\gamma(\sigma)=0$ if $\sigma$ is affine (don't assume this at the outset). You can typically split the geodesic equation up into several differential equations based on what the Christoffel symbols are, and then exploit symmetries to simplify them. Then choose either the $a$ or $b$ coordinate as your parameter $\sigma$, and solve the differential equations.