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

Abstract

We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed in S n x ℝ or ℍ n × 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 S 2 × ℝ or ℍ 2 x R, obtained by rotating the shape operator.