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Isometric immersions into $ \mathbb{S}^n\times\mathbb{R}$ and $ \mathbb{H}^n\times\mathbb{R}$ and applications to minimal surfaces


Author: Benoît Daniel
Journal: Trans. Amer. Math. Soc. 361 (2009), 6255-6282
MSC (2000): Primary 53A10, 53C42; Secondary 53A35, 53B25
DOI: https://doi.org/10.1090/S0002-9947-09-04555-3
Published electronically: July 17, 2009
MathSciNet review: 2538594
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Abstract: We give a necessary and sufficient condition for an $ n$-dimensional Riemannian manifold to be isometrically immersed in $ \mathbb{S}^n\times\mathbb{R}$ or $ \mathbb{H}^n\times\mathbb{R}$ in terms of its first and second fundamental forms and of the projection of the vertical vector field on its tangent plane. We deduce the existence of a one-parameter family of isometric minimal deformations of a given minimal surface in $ \mathbb{S}^2\times\mathbb{R}$ or $ \mathbb{H}^2\times\mathbb{R}$, obtained by rotating the shape operator.


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Additional Information

Benoît Daniel
Affiliation: Institut de Mathématiques de Jussieu, Université Paris 7, Paris, France
Address at time of publication: Département de Mathématiques, Université Paris 12, UFR des Sciences et Technologies, 61 avenue du Général de Gaulle, Bât. P3, 4e étage, 94010 Créteil cedex, France
Email: daniel@univ-paris12.fr

DOI: https://doi.org/10.1090/S0002-9947-09-04555-3
Keywords: Isometric immersion, minimal surface, Gauss and Codazzi equations, integrable distribution
Received by editor(s): May 25, 2007
Published electronically: July 17, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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