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

Abstract

Let $ψ:\M \to \SH$ 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 $λ$ on $\M$ that satisfy the compatibility equations of $\SH$. In this paper, we will deal with the converse problem: "Given a Riemannian manifold $\M$ with symmetric $(1,1)$-tensors $S$ and $f$, tangent vector field $U$ and smooth function $λ$ satisfying the conditions mentioned above, can $\M$ then be isometrically immersed in $\SH$ in such a way that $(g,S,f,U,λ)$ is realized as the induced structure?".