Isometric immersions into 𝕊ⁿ×ℝ and ℍⁿ×ℝ and applications to minimal surfaces

Daniel, Benoît

doi:10.1090/S0002-9947-09-04555-3

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.

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