Uniqueness of Normal Coordinates

Theorem

Let $\struct {M, g}$ be an $n$-dimensional Riemannian or pseudo-Riemannian manifold.

Let $U_p$ be the normal neighborhood for $p \in M$.

Then for every normal coordinate chart on $U_p$ centered at $p \in M$ the coordinate basis is orthonormal at $p$.

Furthermore, for every orthonormal basis $\tuple {b_i}$ for $T_p M$, there is a unique normal coordinate chart $\tuple {x^i}$ on $U_p$ such that:

$\ds \forall i \in \N_{> 0} : i \le n : \valueat{\partial_i} p = b_i$

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Normal Neighborhoods and Normal Coordinates