For connections on trivial vector bundles compatible with compact gauge groups, we establish conditions on the vector bundle and gauge group under which translation of a connection by a constant connection matrix is achievable by a gauge transformation. These conditions may be roughly characterized as either restricting the base manifold to be one-dimensional or restricting the gauge group to take values in an abelian Lie group. These results are then used to prove Poincare inequalities on the gauge equivalent connection matrices, with some additional refinement of these results when the data considered is compactly supported and Coulomb.