Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 4 / c / Solution

In geodesic coordinates built from an orthonormal basis, $g_{ij}(0)={\delta }_{ij}$. Every radial curve has coordinates $x(t)=ta$, and substitution into the geodesic equation givesTaking $t=1$ proves $x^i{x}^j{\Gamma }_{ij}^k(x)=0$.

Conversely, assume the metric and Christoffel-symbol conditions. For every $a$ in the star domain, $x(t)=ta$ satisfies the geodesic equation and has initial velocity $a$ in an orthonormal coordinate frame. Uniqueness of solutions to ordinary differential equations gives ${\varphi }^{-1}(ta)={\exp }_p(ta)$ wherever defined. The star-domain assumption covers all of $U$, so $\varphi$ is precisely a geodesic coordinate chart. This is the Radial Christoffel-symbol criterion for geodesic coordinates.