Properties of reachable sets in the sub-Lorentzian geometry

Abstract

The aim of this paper is to develop local theory of future timelike, nonspacelike and null reachable sets from a given point q 0 in the sub-Lorentzian geometry. In particular, we prove that if U is a normal neighbourhood of q 0 then the three reachable sets, computed relative to U , have identical interiors and boundaries with respect to U . Further, among other things, we show that for Lorentzian metrics on contact distributions on R 2 n + 1 , n ≥ 1 , the boundary of reachable sets from q 0 is, in a neighbourhood of q 0 , made up of null future directed curves starting from q 0 . Every such curve has only a finite number of non-smooth points; smooth pieces of every such curve are Hamiltonian geodesics. For general sub-Lorentzian structures, contrary to the Lorentzian case, timelike curves may appear on the boundary. It turns out that such curves are always Goh curves. We also generalize a classical result on null Lorentzian geodesics: every null future directed Hamiltonian sub-Lorentzian geodesic initiating at q 0 is contained, at least to a certain moment of time, in the boundary of the reachable set from q 0 .