Christoffel Symbols vanish at Origin of Normal Neighborhood

Theorem

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

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

Let $\struct {U_p, \tuple {x^i}}$ be a normal coordinate chart.

Let $\set {\Gamma^i_{jk}}$ be Christoffel symbols.

Then:

$\map {\Gamma^i_{jk} } {\map {x^r}p} = 0$

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

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