The Bonnet problem for surfaces in homogeneous 3-manifolds

We solve the Bonnet problem for surfaces in the homogeneous 3-manifolds with a 4-dimensional isometry group. More specifically, we show that a simply connected real analytic surface in H^2xR or S^2xR is uniquely determined pointwise by its metric and its principal curvatures if and only if it is not a minimal or a properly helicoidal surface. In the remaining three types of homogeneous 3-manifolds, we show that except for constant mean curvature surfaces and helicoidal surfaces, all simply connected real analytic surfaces are pointwise determined by their metric and principal curvatures.