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 $p$.

In geodesic normal coordinates at $p$, radial geodesics have coordinates $x(t)=tv$. The geodesic equation at $t=0$ gives ${\Gamma }_{jk}^i(p)v^j{v}^k=0$ for every $v$. The Levi-Civita connection is torsion free, making these coefficients symmetric in $j,k$; the polarization identity then gives ${\Gamma }_{jk}^i(p)=0$. This does not imply vanishing curvature, which depends on derivatives of the coefficients.

A geodesic sphere is a level set of Riemannian distance from a point. For sufficiently small positive $r$, it is the exponential map image of the radius-$r$ tangent sphere, a smooth compact hypersurface. The Gauss lemma makes its tangent space orthogonal to the radial direction. Larger distance spheres need not be smooth at the cut locus.

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 $t\mapsto tx$.