The fibre bundle of degenerate tangent planes of a Lorentzian manifold and the smoothness of the null sectional curvature

The Grassmann bundle of degenerate tangent planes to a Lorentzian manifold is introduced and several of its properties are deduced. After a suitable normalization, the null sectional curvature associated to a Lorentzian metric may be considered as a well-defined function on the Grassmann bundle of degenerate tangent planes. The smoothness of this function is proved and some consequences are obtained.