Let $(M,g)$ be a Riemannian manifold, denote by $d_g$ the distance on M defined as $$d_g(p,q)=\text{Inf}_{\gamma}\{L[\gamma]\,|\, \gamma:[a,b] \to M, \gamma(a)=p,\, \gamma(b)=q\},$$where $$L[\gamma]=\int_{a}^{b}dt\sqrt{g_{\gamma(t)}(\gamma'(t),\gamma'(t))},$$ for every $p,\,q \in M$.
Fix $p\in M$ and denote $B=B_{d_g}(p,r)$ the ball of center $p$ and radius $r>0$. Then my questions are the following:
Is there a sufficiently small $r>0$ for which there exists a parametrization $(U,x)$ of $M$ satifying that $B \subset x(U) \subset M$?
And, more precisely, is there a sufficiently small $r>0$ for which there exists a parametrization $(U,x)$ of $M$ satifying that $B \subset x(U) \subset M$ and $x^{-1}(B) \subset B_{d_e}(x^{-1}(p),\widetilde{r})\subset U \subset \mathbb{R}^n$ for some $\widetilde{r}>0$ where $d_{e}$ denotes the Euclidean distance in $\mathbb{R}^n$?
It is my undestanding that the natural topology of $(M,g)$ coincides with the topology induced by $d_g$ which somehow connects $d_g$ with the parametrizations $(x,U)$ of $M$. However, it is hard to actually relate $d_g$ and $d_e$ inside a local coordinate system. I am sure you can provide me some general results related to these supposed relations.
Thank you.