Abstract
We study isometric immersions of surfaces of constant curvature into the homogeneous spaces \({\mathbb{H}^2\times\mathbb{R}}\) and \({\mathbb{S}^2\times\mathbb{R}}\) . In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c > 0 into \({\mathbb{H}^2\times\mathbb{R}}\) and a unique one into \({\mathbb{S}^2\times\mathbb{R}}\) 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 \({\mathbb{H}^2\times\mathbb{R}}\) or \({\mathbb{S}^2\times\mathbb{R}}\) .
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J.A. Aledo was partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746 and Junta de Comunidades de Castilla-La Mancha, grant no. PAI-05-034.
J.M. Espinar and J.A. Gálvez were partially supported by Ministerio de Education y Ciencia grant no. MTM2004-02746 and Junta de Andalucía Grant No. FQM325.
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Aledo, J.A., Espinar, J.M. & Gálvez, J.A. Complete surfaces of constant curvature in H2 × R and S2 × R. Calc. Var. 29, 347–363 (2007). https://doi.org/10.1007/s00526-006-0067-4
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DOI: https://doi.org/10.1007/s00526-006-0067-4