Normal coordinates

Remarks

Example: in $\R^n$ with the Euclidean metric and the standard basis, $\exp_p(v) = p + v$, so the normal coordinates centred at $p$ are $\varphi(q) = q - p$, defined on all of $\R^n$.

Non-example: in $\R^2$ with the Euclidean metric, the chart $\psi(q) = 2(q - p)$ has the form $E^{-1} \circ \exp_p^{-1}$ only for the basis $e_i = \tfrac{1}{2} \varepsilon_i$, where $\varepsilon_1, \varepsilon_2$ is the standard basis. This basis is orthogonal but not orthonormal, so $\psi$ is not a normal coordinate chart; indeed, in $\psi$ the metric has matrix $\tfrac{1}{4} I$ at $p$, not $I$.

In normal coordinates centred at $p$: $g_{ij}(p) = \delta_{ij}$, $\partial_k g_{ij}(p) = 0$ and $\Gamma^k_{ij}(p) = 0$. The geodesics through $p$ are the lines $t \mapsto t x$ in these coordinates. Different orthonormal bases give normal coordinates that differ by an orthogonal matrix.