Hypersurfaces and Codazzi tensors

Abstract.

In this paper we deal with the following problem. Let (M ⁿ,〈,〉) be an n-dimensional Riemannian manifold and $f:(M^{n},\langle \,,\rangle)\rightarrow {\Bbb R}^{n+1}$ an isometric immersion. Find all Riemannian metrics on M ⁿ that can be realized isometrically as immersed hypersurfaces in the Euclidean space ${\Bbb R}^{n+1}$. More precisely, given another Riemannian metric $\widetilde{{\langle \,,\rangle }}$ on M ⁿ, find necessary and sufficient conditions such that the Riemannian manifold $(M^{n},\widetilde{{\langle \,,\rangle}})$ admits an isometric immersion ${\tilde{f}}$ into the Euclidean space ${\Bbb R}^{n+1}$. If such an isometric immersion exists, how can one describe ${\tilde{f}}$ in terms of f?