User anon - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T05:35:26Z http://mathoverflow.net/feeds/user/8090 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/34088/a-geometric-interpretation-of-the-levi-civita-connection/34133#34133 Answer by anon for A geometric interpretation of the Levi-Civita connection? anon 2010-08-01T18:20:22Z 2010-08-01T18:20:22Z <p>Is the following description correct?</p> <p>The metric determines the geodesics: pull a string tight enough and it will be a geodesic.</p> <p>These in turn determine a class of connections, determined up to torsion: twist the string while parallel transporting a tangent vector along it, and you are changing the connection keeping the same geodesics.</p> <p>Now choose a connection, parallel transport an infinitesimal vector along a geodesic curve $\gamma$. The tip of the vector will draw a curve $\gamma '$. The zero torsion connection in the class, i.e. the Levi-Civita connection, is the one minimizing the lenght of $\gamma '$. </p> <p>BTW, this question is related: <a href="http://mathoverflow.net/questions/20493" rel="nofollow">http://mathoverflow.net/questions/20493</a></p>