Conformally flat Lorentzian hypersurfaces in Lorentzian 4-space with special shape operator

In the conformal (Möbius) geometry of submanifolds, using algebraic invariants of the shape operator to construct conformal invariants is a frequently used method. But it does not apply to 3-dim Lorentzian hypersurfaces of the last type, the minimal polynomial of whose shape operator has a triple root. In this paper, using the obstruction of some distribution to be integrable, a new method to construct conformal invariants is introduced. Using this method, a complete conformal invariant system is constructed for generic conformally flat Lorentzian hypersurfaces of the last type. We find that such kind of hypersurfaces allows an infinite parameter family of non-equivalent deformations, which implies they are more abundant than the other three types. For non-generic conformally flat Lorentzian hypersurfaces of the last type, the isometric geometry is studied and a fundamental theorem is obtained in this paper.