Conformally flat Lorentz hypersurfaces

Abstract

We shall investigate conformally flat Lorentz hypersurfaces in indefinite space forms. Some particular classes of such hypersurfaces are explicitly described and classified.

Introduction

Recall that a pseudo-Riemannian manifold (M,g) is said to be conformally flat if each x∈M belongs to a neighborhood U⊂M such that, for certain σ∈C^∞(U), the submanifold $\text{(U,}\text{e}{}^{\mathrm{\sigma }}\text{g)}$ is flat. Nonflat conformally flat Riemannian hypersurfaces in Euclidean spaces Eⁿ⁺¹, n≥4, had been firstly investigated by Cartan [3], who showed that the second fundamental form of those hypersurfaces admits at each point an eigenvalue of multiplicity ≥n−1. Conformally flat Riemannian hypersurfaces in positive definite space forms had been extensively studied by Chen (cf. [4]), and classified by Do Carmo et al. in the compact case (cf. [6]).

In this paper, we deal with conformally flat Lorentz hypersurfaces of dimension n≥4 in indefinite space forms $\text{M}\text{̃}{}^{\mathrm{n+1}}\text{(}\text{c}\text{̃}\text{)}$, i.e., complete simply connected and connected (n+1)-dimensional Lorentz manifolds of constant curvature $\text{c}\text{̃}$. In case $\text{M}\text{̃}{}^{\mathrm{n+1}}\text{(}\text{c}\text{̃}\text{)=}\text{R}{}_1{}^{\mathrm{n+1}}$, n≥4, a local classification of these hypersurfaces was obtained by Van de Woestijne and Verstraelen (see [19, Theorem 2]). They claimed that, if the induced metric on a Lorentz hypersurface $\text{M}{}^{\mathrm{n}}\text{⊂}\text{R}{}_1{}^{\mathrm{n+1}}$ is conformally flat, then it can be described as follows. Locally, Mⁿ is either congruent to a part of a hypercylinder, a Lorentz hypersphere, a generalized cylinder, or a generalized umbilical hypersurface; or else Mⁿ is foliated by (n−1)-dimensional Euclidean or Lorentzian hyperspheres, paraboloids, or hyperbolic spaces. Those hypersurfaces which are foliated by paraboloids or hyperbolic spaces would consist only of what we will call “bad points”. However, it should be remarked that while Theorem 2 in [18] may be correct, its proof do not seem to be clear (one should actually provide evidence of the argument used in that paper and which consists in the fact that if the shape operator A takes a certain form at a point x∈Mⁿ, then the same form still holds in a neighborhood of x and the eigenvalues of A have constant multiplicities in that neighborhood).

The paper is organized as follows. Section 2 contains the basic facts about general hypersurfaces in space forms. It also contains notation and formulas we will be using. Section 3 contains a classification result for shape operators of conformally flat Lorentz hypersurfaces in space forms. Section 4 presents the standard examples of Lorentz hypersurfaces which will serve as models in our classification. Section 5 is the main section, it contains various results on conformally flat Lorentz hypersurfaces in indefinite space forms in general. For instance, those hypersurfaces in Minkowski space which we call “good hypersurfaces” are explicitly described and classified.