Are the normal coordinates of a Riemannian manifold compatible with its original charts?

Question

As above, any Riemannian manifold can be equipped with normal coordinates. My question is whether those normal coordinates are compatible with its original coordinates charts?

Answer 1

Yes, otherwise they wouldn't be called normal coordinates. This follows from the fact that $\exp_p$ (restricted to an appropriate neighborhood) is a diffeomorphism and so is $\varphi$. Any diffeomorphism $\varphi \colon U \rightarrow \mathbb{R}^n$ from an open subset of $M$ to an open subset of $\mathbb{R}^n$ defines a coordinate system $x^1,\dots,x^n$ on $U$ (where $\varphi = (x^1,\dots,x^n)$). The fact that $\varphi$ is a diffeomorphism (with respect to the natural smooth structure on $U$ as an open subset of $M$ and the natural smooth structure on $\mathbb{R}^n$) guarantees that the coordinates are indeed compatible with the smooth structure on $M$.