Abstract
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 H2×R or S2×R 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 3manifolds, 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.
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