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The Bonnet problem for surfaces in homogeneous 3-manifolds

Jose A. Galvez·Antonio Martinez·Pablo Mira
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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 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.

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Jose A. Galvez, Antonio Martinez, Pablo Mira. 2006-12-26. The Bonnet problem for surfaces in homogeneous 3-manifolds. https://arxiv.org/abs/math/0612766

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