We classify SU(2)-cyclic and SU(2)-abelian 3-manifolds, for which every representation of the fundamental group into SU(2) has cyclic or abelian image, respectively, among geometric 3-manifolds that are not hyperbolic. As an application, we give examples of hyperbolic 3-manifolds that do not admit degree-1 maps to any Seifert Fibered manifold other than S3 or a lens space. We also produce infinitely many one-cusped hyperbolic manifolds with at least four SU(2)-cyclic Dehn fillings, one more than the number of cyclic fillings allowed by the cyclic surgery theorem.