On a theorem of Preissmann

Abstract

PROOF. Let xoEM and o CM such that 7r(xo) =xo. Let Z be a path in M joining xo to a(xo) and let C be its projection into M. According to Bishop and Crittenden [1, p. 293] there is a closed geodesic in every free homotopy class of loops on M. Let C1 be a closed geodesic which is free homotopic to C and let F: [0, 1 ] X [0, 1 ]--M be the free homotopy joining C to C1, i.e. F(x, O)=C(x), F(x, 1) =C1(x), F(O, t) = F(1, t). Let G(t) = F(O, t) and let G be the unique lift of this path to M such that G(O) =xo. Let G(1) =yo. It follows from elementary properties of covering spaces that the loop C1 and the base point yo induce the transformation a. However, the former transformation obviously preserves the geodesic in M which passes through yo and covers C1. Q.E.D.