Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant   Riemannian metrics
  We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of variations and classical analysis only. The geodesic distance is discussed for some special cases and applications towards the theory of nonlinear elasticity are indicated. 
Robert Martin and Patrizio Neff
math.DG