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

TL;DR

This paper investigates the conditions under which a Riemannian manifold with certain tensor fields can be isometrically immersed into a product of a sphere and hyperbolic space, focusing on the converse problem of compatibility equations.

Contribution

It provides a characterization of when a Riemannian manifold with specified tensors and functions can be immersed into the product space, extending the understanding of hypersurfaces in such ambient spaces.

Findings

Established necessary and sufficient conditions for isometric immersion.

Extended classical results to the setting of product spaces.

Provided a framework for constructing hypersurfaces with prescribed geometric data.

Abstract

Let $\psi:\M \to \SH$ 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 $\lambda$ 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 $\lambda$ satisfying the conditions mentioned above, can $\M$ then be isometrically immersed in $\SH$ in such a way that $(g,S,f,U,\lambda)$ is realized as the induced structure?".