  We give a necessary and sufficient condition for a 2-dimensional Riemannian
manifold to be locally isometrically immersed into a 3-dimensional homogeneous
manifold with a 4-dimensional isometry group. The condition is expressed in
terms of the metric, the second fundamental form, and data arising from an
ambient Killing field. This class of 3-manifolds includes in particular the
Berger spheres, the Heisenberg space Nil(3), the universal cover of the Lie
group PSL(2,R) and the product spaces S^2 x R and H^2 x R. We give some
applications to constant mean curvature (CMC) surfaces in these manifolds; in
particular we prove the existence of a generalized Lawson correspondence, i.e.,
a local isometric correspondence between CMC surfaces in homogeneous
3-manifolds.
