Semi-Riemannian hypersurfaces in $$\mathbb {L}^{n+1}$$ L n + 1 with a totally geodesic foliation of codimension one

Abstract

We classify hypersurfaces of the Minkowski space $$\mathbb {L}^{n+1}$$ that carry a totally geodesic foliation with complete leaves of codimension one. We prove that such a hypersurface is ruled, or a partial tube over a curve or contains a two or three dimensional strip. Moreover, if the hypersurface is embedded then it is a partial tube over a curve.