Isometric immersions of higher codimension into the product \(S^k {\times} H^{n+p-k}\) (Q477928)

The authors obtain a necessary and sufficient condition for a simply connected Riemannian manifold \((M^n,g)\) to be isometrically immersed, as a submanifold with codimension \(p\geq 1\), into the Riemannian product \(S^k\times H^{n+p-k}\) of a sphere and a hyperboloid. By a minor modification, one can prove that the main theorem remains true if one replaces the ambient product space by a direct product \(M^k(c_1)\times M^{n+p-k}(c_2)\) of any two real space forms with constant curvatures \(c_1\) and \(c_2\), respectively.