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 ?



Minimal representative of the elements of the fundamental group of a negatively curved manifold
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...


