Sub-Lorentzian Structures in ℝ 4 $\mathbb {R}^{4}$ : Left-Invariance and Conformal Normal Forms

Abstract

Germs of sub-Lorentzian structures on hyperplane distributions in four-dimensional space are divided into three types: elliptic, hyperbolic, and parabolic (provided the distribution satisfies some non-integrability condition). Their main quasi-homogeneous parts are non-integrable left-invariant sub-Lorentzian structures on the trivially extended Heisenberg group, and there are exactly three such structures up to isomorphisms of the group: elliptic, hyperbolic, and parabolic. For elliptic and hyperbolic germs, we normalize their next-order quasi-homogeneous terms up to conformal equivalence. A control-affine system in three-dimensional space with a disk of admissible velocities defines a sub-Lorentzian structure in the four-dimensional space-time. If the latter is elliptic and left-invariant up to conformal equivalence, then the attainable sets of the original control-affine system are diffeomorphic to the well-known sub-Riemannian ball in the Heisenberg group. The attainable sets in the hyperbolic case are described in the present paper.