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I have the following cuestion: Is every Riemannian metric "locally" flat?, i.e., for every poit $p\in M$ there is a smooth chart around $p$ such that the Christoffel symbols are all zero.

If this is true, how can i obtain such chart?

Thanks in advance.

kahlerian
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    You are misusing the language. "At a single point" is NOT "locally." Yes, you can make the Christoffel symbols be zero at a point, but not locally. So you know nothing about curvature. – Ted Shifrin Feb 15 '21 at 22:06

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This is not true as the curvature is $C^2$-invariant and is determined by the Christoffel symbols.

It is possible to prescribe the metric at a point (this is a matter of linear algebra), but not in an open neighbourhood.

C. Falcon
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  • I guess this is a duplicate of this https://math.stackexchange.com/questions/40931/normal-coordinates-vs-locally-flat?rq=1 – kahlerian Feb 15 '21 at 22:05