Fibered representations, an open condition

Let M be a closed (connected) smooth manifold and G be a Lie group. Denote by R0(π1M,G) the space of all representations φ:π1(M)→G such that A=image(φ) is a discrete and torsion-free subgroup of G. We show that an open subspace of R0(π1M,G) consists of all those φ which are determined by a fibre bundle projection p:M→A\G/K. Here K is a maximal compact subgroup of G and “determined by p” means that φ=p#. (There are some restrictions on G for our proof to work.) This generalizes a well-known result of Tischler which covers the case where G=R. In his situation the double coset space A\G/K is always the circle.