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?
The exp map and distance on Riemannian manifolds
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riemannian-geometry
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1The 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. – 2011-07-16
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2[See the first bullet point here](http://en.wikipedia.org/wiki/Normal_coordinates#Properties). @Sam: you could have said that the norm is the square root of the quantity the OP asks about :) – 2011-07-16
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1@Sam: ideally, you add a reference to wrap it up, and you can turn that comment into an answer! :) – 2011-07-17
1 Answers
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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.