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If $(M, g)$ be a Riemannian manifold. My doubt is the following:

For each $p\in M$, there is a coordinate chart $(U, (x_1,x_2,.., x_n))$ such that $g = \sum dx_i\otimes dx_j $

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No. Locally, you can always find an orthonormal frame $\{X_1\ldots X_n\}$, but its vectors need not be coordinate vectors relative to a chart, because in general you will have $[X_i, X_j]\ne 0$. The property you state is equivalent to the manifold being flat.

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