MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have a simple question regarding the differentiation of the logarithm mapping in Riemannian manifolds:

Assume that $M$ is a compact Riemannian manifold, isometrically embedded into $\mathbb{R}^n$.

Let $p(t), q(t): [0,1] \to M$ be two curves.

I am interested in bounding the quantity

$$ \frac{d}{dt}\log_{p(t)}(q(t)), $$

more precisely, does the estimate $$ \left|\frac{d}{dt}\log_{p(t)}(q(t))\right|\lesssim \left|\frac{d}{dt}p(t) - \frac{d}{dt}q(t)\right| + \left|p(t) - q(t)\right| $$ hold true, at least if $p(t)$ and $q(t)$ are close to each other??

share|cite|improve this question
1) Why do you need to isometrically embed the manifold into $\mathbb{R}^n$? 2) Your question is best posed in terms of Jacobi fields. I think if you take the trouble to do that, you might be able to figure out whether your inequality or something like it holds. – Deane Yang Nov 1 '12 at 1:24
thanks for your comment. 1) because otherwise the right hand side would not make sense. 2) thanks for the suggestion of using Jacobi fields. I tried it and die not see whether the inequality holds... – pil Nov 1 '12 at 7:23

Your Answer


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

Browse other questions tagged or ask your own question.