Hypersurfaces M^n in S^k x H^n-k+1

Let $\psi :\text{M}\to \text{SH}$ be an isometric immersion of codimension 1, then there exist symmetric $(1,1)$-tensors $S$ and $f$, a tangent vector field $U$ and a smooth function $\lambda$ on $\text{M}$ that satisfy the compatibility equations of $\text{SH}$. In this paper, we will deal with the converse problem: “Given a Riemannian manifold $\text{M}$ with symmetric $(1,1)$-tensors $S$ and $f$, tangent vector field $U$ and smooth function $\lambda$ satisfying the conditions mentioned above, can $\text{M}$ then be isometrically immersed in $\text{SH}$ in such a way that $(g,S,f,U,\lambda )$ is realized as the induced structure?”.