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 ?