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?
All you need to do is find the Christoffel symbols, defined by
$$\Gamma_{\rho\tau}^\mu=\frac{1}{2}g^{\mu\nu}\left(\frac{\partial g_{\nu\rho}}{\partial x^\tau}+\frac{\partial g_{\nu\tau}}{\partial x^\rho}-\frac{\partial g_{\rho\tau}}{\partial x^\nu}\right)$$
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.
