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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 ?

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    $\begingroup$ This question is more appropriate for math.stackexchange.com. I suggest you migrate it to there. $\endgroup$
    – Deane Yang
    Apr 18 '19 at 14:47
  • $\begingroup$ Or try to answer this yourself using the fact that you know exactly what the geodesics are for a flat manifold. $\endgroup$
    – Deane Yang
    Apr 18 '19 at 18:49
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What you have described are indeed Cartesian coordinates on a flat space form.

However, this is not the usual definition of normal coordinates, which instead take the exponential of a vector without decomposing it in terms of an orthonormal basis. On a flat space form (i.e. Euclidean space), the standard definition of normal coordinates actually correspond to spherical coordinates.

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