Metric is locally constant 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 $


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.
