Communications in Analysis and Geometry

Volume 16 (2008)

Number 5

The Bonnet problem for surfaces in homogeneous $3$-manifolds

Pages: 907 – 935

DOI: http://dx.doi.org/10.4310/CAG.2008.v16.n5.a1

Authors

José A. Gálvez (Departamento de Geometría y Topología, Universidad de Granada)

Antonio Martínez (Departamento de Geometría y Topología, Universidad de Granada)

Pablo Mira (Universidad Politécnica de Cartagena)

Abstract

We solve the Bonnet problem for surfaces in the homogeneous\hbox{$3$-manifolds} with a $4$-dimensional isometry group.More specifically, we show that a simply connectedreal-analytic surface in $\H^2\times \R$ or $\S^2\times \R$is uniquely determined pointwise by its metric and itsprincipal curvatures if and only if it is not a minimal ora properly helicoidal surface. In the remaining three typesof homogeneous $3$-manifolds, we show that except forconstant mean curvature surfaces and helicoidal surfaces,all simply connected real-analytic surfaces are pointwisedetermined by their metric and principal curvatures.

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