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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??

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  • $\begingroup$ 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. $\endgroup$
    – Deane Yang
    Commented Nov 1, 2012 at 1:24
  • $\begingroup$ 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... $\endgroup$
    – pil
    Commented Nov 1, 2012 at 7:23
  • $\begingroup$ Did you ever figure this out? $\endgroup$
    – ABIM
    Commented Jun 30, 2018 at 15:28

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