Causal Invariance of the Contact Number of Pseudo-Riemannian Submanifolds

Abstract

Let $M_s^n$ be a non-degenerate pseudo-Riemannian submanifold of a pseudo-Euclidean space with indefinite induced metric. Restricting the contact condition to spacelike or to timelike tangent directions gives two a priori different one-sided contact numbers. We prove that, for every prescribed order, ordinary, spacelike and timelike contact are equivalent; consequently, the three contact numbers coincide. The proof is inductive. A one-sided contact hypothesis first yields higher-order orthogonality identities for the iterated covariant derivatives of the second fundamental form. Their diagonal scalar expressions are then extended from either unit pseudo-sphere to the whole tangent space by a polynomial argument, and a second induction proves causal invariance. We also construct an explicit Lorentzian surface in $\mathbb E_4^8$ whose ordinary, spacelike and timelike contact numbers are all equal to six, and place the infinite-contact limit in the semi-Riemannian theory of geodesic normal sections and helical immersions.