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> gives
$$
a^ia^j\Gamma^k_{ij}(ta)=0.
$$
Taking $t=1$ proves $x^ix^j\Gamma^k_{ij}(x)=0$.

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 $\phi^{-1}(ta)=\exp_p(ta)$ wherever defined. The star-domain assumption covers all of $U$, so $\phi$ is precisely a geodesic coordinate chart. This is the <Radial Christoffel-symbol criterion for geodesic coordinates>.

Solution

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

{c}

A chart $x$ centered at $p$ on a star-shaped coordinate domain is a geodesic normal chart exactly when $g_{ij}(0)=\delta_{ij}$ and
$$
x^ix^j\Gamma^k_{ij}(x)=0
$$
for every $x$ and $k$. This is the geodesic equation for every radial curve $t\mapsto tx$.


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 Incoming links: Radial Christoffel-symbol criterion for geodesic coordinates