Conformally flat Lorentz hypersurfaces. (Q1400160)

The conformally flat Lorentz hypersurfaces \(M^n\) of dimension \(n\geq 4\) in indefinite space forms \(\widetilde M^{n+1}(\widetilde c)\) are investigated. Here \(\widetilde M^{n+1}(\widetilde c)\) is a complete simply connected \((n+1)\)-dimensional Lorentz manifold of constant curvature \(\widetilde c\). In case \(\widetilde M^{n+1}(\widetilde c)= \mathbb{R}^{n+1}_1\) these hypersurfaces are locally classified by \textit{I. van de Woestijne} and \textit{L. Verstraelen} [Tôhoku Math. J., II. Ser. 39, 81--88 (1987; Zbl 0627.53042)]. In the present paper first a classification of shape operators \(A_r\) of conformally flat Lorentz hypersurfaces \(M^n\) for \(x\in M^n\) in space forms is given. Then the standard examples of Lorentz hypersurfaces are presented as models for the further classification. If \(A_x\) is nonsingular and has a simple eigenvalue, the point \(x\in M^n\) is called bad, all other points are called good. If all points are good, then \(M^n\) is said to be good. In the main result all conformally flat good Lorentz hypersurfaces \(M^n\) for \(n\geq 4\) in \(M^{n+1}(c)\) are explicitly described and classified. For the bad points it is established that if \(M^n\) of scalar curvature \(r\), \(n\geq 4\), is a conformally flat Lorentz hypersurface in \(\widetilde M^{n+1}(\widetilde c)\) with \(\widetilde c> r/n(n-1)\) everywhere, then all points are bad, and \(M^n\) is foliated by \((n-1)\)-dimensional spaces of constant curvature \(>\widetilde c\).