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-2 x R or S-2 x 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 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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机译：我们用4维等距组解决了齐次3流形中曲面的Bonnet问题。更具体地说，我们表明，当且仅当它不是最小曲面或适当的螺旋曲面时，H-2 x R或S-2 x R中简单连接的实解析面是由其度量和主曲率唯一地逐点确定的。在其余三种均质的3流形中，我们表明，除了恒定的平均曲率曲面和螺旋曲面之外，所有简单连接的实解析面都是通过其度量和主曲率逐点确定的。
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