Geodesic normal coordinates

Choose an orthonormal basis of $T_{p}M$ and use its linear coordinates on a neighborhood of zero on which ${\exp }_p$ is a diffeomorphism. Transporting these coordinates through ${\exp }_p$ gives geodesic normal coordinates centered at $p$.

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