most recent 30 from http://mathoverflow.net 2013-05-22T02:49:53Z http://mathoverflow.net/feeds/question/111070 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111070/differentiation-of-logarithm-map-in-riemannian-geometry pil 2012-10-30T10:49:11Z 2012-10-30T10:49:11Z

<p>I have a simple question regarding the differentiation of the logarithm mapping in Riemannian manifolds:</p> <p>Assume that $M$ is a compact Riemannian manifold, isometrically embedded into $\mathbb{R}^n$. </p> <p>Let $p(t), q(t): [0,1] \to M$ be two curves. </p> <p>I am interested in bounding the quantity</p> <p>$$ \frac{d}{dt}\log_{p(t)}(q(t)), $$</p> <p>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??</p>