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

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.