Isometric immersions of higher codimension into the product ${\mathbf{S}}^k\times H^{n+p-k}$ of sphere and hyperboloid

In this paper, we obtain a sufficient and necessary condition for a simply connected Riemannian manifold $(M^n,g)$ to be isometrically immersed, as a submanifold with codimension $p\ge 1$, into the product $S^k\times H^{n+p-k}$ of sphere and hyperboloid.