On the regularity of null cones and geodesic spheres (Q2697621)

It is wellknown that the exponential map in a Riemannian manifold is never injective near a conjugate point. To study similar properties on the behaviour of the exponential map in a Lorentzian or Riemannian manifold, one lookes at null hypersurfaces. These are hypersurface in a Lorentzian manifold such that the inherited metric is degenerated. On the other hand, a null cone with vertex at a point \(p_0\) in a Lorentzian manifold is the set formed by null geodesics emanating from \(p_0\) such that their initial velocities are in the same timecone than a fixed timelike vector \(e_0\). It is a question whether a null cone, regarded as a subset of a Lorentzian manifold, be considered as a smooth (null) hypersurface despite the presence of null conjugate points. In the present paper the authors give a negative answer to this question. In fact, they show that there is no neighbourhood of a singular point in the null cone of the exponential map which image is contained in a null hypersurface.