Isometric immersions into \(\mathbb{S}^m \times \mathbb{R}\) and \(\mathbb{H}^m \times \mathbb{R}\) with high codimensions (Q977713)

\textit{B. Daniel} [Trans. Am. Math. Soc. 361, No.~12, 6255--6282 (2009; Zbl 1213.53075)], gave a necessary and sufficient condition for an \(n\)-dimensional Riemannian manifold to be isometrically immersed in \(S^n \times {\mathbb R}\) or \(H^n \times {\mathbb R}\) in terms of its first and second fundamental forms and some other conditions. In the present paper the authors generalise this to higher codimensions. The necessary and sufficient conditions are the three equations of Gauss, Codazzi and Ricci, together with two conditions on the vector field on \(M\) which is obtained by projection of the vector field \(\partial/\partial t\) associated to the \({\mathbb R}\)-factor in the target space.