  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 which are not hyperbolic. As an
application, we give examples of hyperbolic 3-manifolds which do not admit
degree-1 maps to any Seifert fibered manifold other than $S^3$ 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.
