Complete surfaces of constant curvature in H2xR and S2xR

We study isometric immersions of surfaces of constant curvature into the homogeneous spaces H2xR and S2xR. In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c>0 into H2xR and a unique one into S2xR when c>1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c<-1 cannot be isometrically immersed into H2xR or S2xR.