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Let $M$ be Riemann manifold,would it possible to find a local orthonormal frame around $p$ denote it $(E_i)$ such that at point $p$ we have $\nabla_{E_i}E_j = 0$?

My attempt,first we can find a orthonormal frame around point $p$,second we can find a normal coordinate ,we know that $\Gamma^k_{i,j}$ vanish at point $p$.hence we have $\nabla_{\partial_i} \partial_j = 0$ at point $p$.Finally we can take rotation such that $\partial_i = E_i$ at point $p$.

The problem here is $\nabla_{E_i}E_j = \nabla_{\partial_i}E_j \ne \nabla_{\partial_i}\partial _j$ typically ,so would it possible to choose some orthonormal frame such that $\nabla_{E_i}E_j = 0$ at point $p$?

yi li
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  • What happens if you apply the Gram-Schmidt orthogonalization procedure to $\partial_i$'s? – Michał Miśkiewicz Sep 08 '21 at 10:12
  • $\partial_i$ are already orthonormal at point $p $ – yi li Sep 08 '21 at 10:14
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    But they're not orthonormal in the whole neighborhood. That's the point, right? Otherwise we could take $E_i := \partial_i$. So instead, we can take $E_i$ to be the outcome of GS (applied pointwise in the neighborhood). – Michał Miśkiewicz Sep 08 '21 at 10:15
  • there is another question how can we guarantee that $\nabla_{E_i}E_j = 0$? the problem now is $\nabla_{\partial_i}E_j = 0$ at $p$? @Michał Miśkiewicz – yi li Sep 08 '21 at 10:46
  • Yep, there's a question here, that's why it's a comment and not an ansewr. But at least it's a very concrete thing to try out. – Michał Miśkiewicz Sep 08 '21 at 11:18
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    thank you I found a post here https://math.stackexchange.com/q/109917/360262 which gives a idea using parallel transport along the radial geodesics inside the normal coordinate . – yi li Sep 08 '21 at 11:19
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    Do you know the basics of moving frames and working with the connection $1$-forms and how they transform when you change frame? This approach is very efficient for answering your question. – Ted Shifrin Sep 08 '21 at 16:08
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    @TedShifrin, not just efficient but also much simpler. There many arguments like this, where people use the exponential map but really need only a simple calculation using a change of coordinates or change of frame up to first or second order at a point. – Deane Sep 08 '21 at 16:45
  • @Ted Shifrin thank you, I don't know this type of argument, can you provide some reference? – yi li Sep 09 '21 at 00:56
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    Any good differential geometry book that treats moving frames and connections on principal bundles. Chern/Chen/Lam and volumes 2 and 3 of Spivak are good examples. More elementary is Jeanne Clelland’s new book. – Ted Shifrin Sep 09 '21 at 01:04

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