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Given a Riemannian manifold $(M, g)$ and $p \in M$, is it possible to find a local orthonormal frame about $p$? i.e.

in $U \ni p$, there exists $v_1, \cdots, v_n \in TU$ such that $g(v_i, v_j) = \delta_{ij}$.

If I want to find a set of orthonormal tangent vectors $v_1|_p, \cdots, v_n|_p$ at $T_pM$, then it is merely Gram-Schmidt. However, I am wondering if I can do the same in an arbitrarily small neighborhood. Is it possible to apply Gram-Schmidt on $U$?

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    $\begingroup$ Yes, Gram-Schmidt with parameter works: $v_1(p)=e_1/\sqrt{g(p)(e_1,e_1)}$, $v_2(p)=[e_2-g(p)(e_2,v_1)v_1]/\sqrt{\dots}$, etc. $\endgroup$ Commented Apr 6, 2021 at 14:41
  • $\begingroup$ I think it is even possible to find a transformation of the local coordinates, such that these coordinates induce such a orthonormal base. $\endgroup$
    – user408858
    Commented Apr 6, 2021 at 14:43
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    $\begingroup$ @user408858, no. There are local orthonormal coordinates if and only if the metric is flat. $\endgroup$
    – Deane
    Commented Apr 6, 2021 at 15:36
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    $\begingroup$ Your question is very similar to this one from yesterday. I suggest you to check my answer on this. $\endgroup$
    – Didier
    Commented Apr 6, 2021 at 15:40

1 Answer 1

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Given a Riemannian manifold $M$ we know by Levi-Civita that there exists a unique affine connection $\nabla$ on $M$ such that it is compatible with the metric and symmetric. Remember compatibility means that the inner product is preserved under parallel transport along smooth curves. So in our case, start at $p\in M$ and fix and ONB $\{v_1,...,v_n\}$. We may the consider all smooth curves passing through $p$ and see that $\langle v_i,v_j\rangle=\delta_{ij}$ along all such curves.

I am by no means an expert on differential geometry but I think it's important to point out that this is different that viewing the metric as the Euclidean metric in a neighborhood. This in general cannot be done without higher order corrections and is intimately related to the curvature of the manifold.

Check out Riemann Normal Coordinates. This tells us that we can find a coordinate system in which the metric is Euclidean around any point but with higher order corrections ie $$g_{ij}(x_1,x_2)=\delta_{ij}-\frac{1}{3}\sum_l \sum_k R_{iljk}x_1^l x_2^k +\mathcal{O}(|x|^2)$$ where $R_{iljk}$ are the components of the Riemann curvature tensor in this coordinate system.

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