Sub-Lorentzian Structures in $R^4$: Left-Invariance and Conformal Normal FormsстатьяИсследовательская статья
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Дата последнего поиска статьи во внешних источниках: 4 сентября 2018 г.
Аннотация: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 conditi on). 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.