Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
1  
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
2  
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
1  
@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
add comment

1 Answer 1

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.