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Let $(M,g)$ be a Riemannian manifold. Let $p$ be a point in $M$, and suppose we create a diffeomorphism between the tangent space at $p$ and a small neighborhood of $p$ in $M$. Is it then true that the distance between $q$ and $p$ is $\langle v,v\rangle$, where $v=p-q$ in the tangent space, where we use the exp map to locate $q$ in the tangent space?

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The distance $d(p,q)$ will be the norm $|v|_g$ of $v$ with $\exp_p(v) = q$ (under the assumption that the neighborhood is chosen small enough). But this can be found in any book on Riemannian geometry. – Sam Jul 16 '11 at 23:31
See the first bullet point here. @Sam: you could have said that the norm is the square root of the quantity the OP asks about :) – t.b. Jul 16 '11 at 23:43
@Sam: ideally, you add a reference to wrap it up, and you can turn that comment into an answer! :) – Mariano Suárez-Alvarez Jul 17 '11 at 0:35
up vote 2 down vote accepted

Just to give a reference for "any book on Riemannian geometry": A proof of the above (and much more) can be found in do Carmo's Riemannian Geometry.

Your question is answered in chapter 3; In particular paragraph 3 of this chapter treats minimizing properties of geodesics.

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