Isometric immersions into $\mathbb {S}^n\times \mathbb {R}$ and $\mathbb {H}^n\times \mathbb {R}$ and applications to minimal surfaces
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- by Benoît Daniel PDF
- Trans. Amer. Math. Soc. 361 (2009), 6255-6282 Request permission
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.References
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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
- Received by editor(s): May 25, 2007
- Published electronically: July 17, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2538594