Hypersurfaces with Light-Like Points in a Lorentzian Manifold

Abstract

Consider a constant mean curvature immersion $$F:U(\subset \varvec{R}^n)\rightarrow M$$ into an arbitrary Lorentzian $$(n+1)$$-manifold M. A point $$o\in U$$ is called a light-like point if the first fundamental form $$\mathrm{d}s^2$$ of F degenerates at o. We denote by $$B_F$$ the determinant function of the symmetric matrix associated to $$\mathrm{d}s^2$$ with respect to a local coordinate system at o. A light-like point o is said to be degenerate if the exterior derivative of $$B_F$$ vanishes at o. We show that if o is a degenerate light-like point, then the image of F contains a light-like geodesic segment of M passing through f(o) (cf. Theorem E). This explains why several known examples of constant mean curvature hypersurface in the Lorentz–Minkowski $$(n+1)$$-space form $$\varvec{R}^{n+1}_1$$ contain light-like lines on their sets of light-like points, under a suitable regularity condition of F. Several related results are also given.