Are normal coordinates the same as Cartesian coordinates in flat space? [closed]

Let $$\gamma_v$$ be the unique maximal geodesic with initial conditions $$\gamma_v(0)=p$$ and $$\gamma_v'(0)=v$$ then the exponential map is defined by
$$\exp_p(v)=\gamma_v(1)$$

If we pick any orthonormal basis, $$(e_1,\ldots,e_n)$$ of $$T_pM$$ then the $$x_i$$’s, with $$x_i = \operatorname{pr}_i \circ \exp^{-1}$$ and $$\operatorname{pr}_i$$ the projection onto $$Re_i$$ , are called normal coordinates at $$p$$.

My question is , if our manifold $$M$$ is flat than does this normal coordinates coincide to what we call Cartesian coordinates ?

• This question is more appropriate for math.stackexchange.com. I suggest you migrate it to there. Apr 18 '19 at 14:47
• Or try to answer this yourself using the fact that you know exactly what the geodesics are for a flat manifold. Apr 18 '19 at 18:49