Minimal representative of the elements of the fundamental group of a negatively curved manifold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T23:53:19Z http://mathoverflow.net/feeds/question/81877 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81877/minimal-representative-of-the-elements-of-the-fundamental-group-of-a-negatively-c Minimal representative of the elements of the fundamental group of a negatively curved manifold student 2011-11-25T12:40:34Z 2011-11-25T14:05:28Z <p>Let (M,g) be a negatively curved manifold , let p be any point of M and denote by G=π1(M,p) . the minimal representative (by minimal i mean the smallest length representative ) of every α in G is a simple closed geodesic loop at p . my question is why it should be simple ?</p> http://mathoverflow.net/questions/81877/minimal-representative-of-the-elements-of-the-fundamental-group-of-a-negatively-c/81886#81886 Answer by Igor Rivin for Minimal representative of the elements of the fundamental group of a negatively curved manifold Igor Rivin 2011-11-25T14:05:28Z 2011-11-25T14:05:28Z <p>For a negatively curved manifold, there is a unique geodesic in a free homotopy class, and a unique geodesic broken loop in a homotopy class, and it is the shortest curve. In neither case is the curve necessarily simple. For references, almost any book on differential geometry will work (Cheeger/Ebin, Ballmann/Gromov/Schroeder, Bridson/Haefliger are all good candidates for having a discussion). For curves on surfaces, you might want to check out the little paper of McShane/Rivin in IMRN, which talks about minimal representatives in homology classes...</p>