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Sub-Lorentzian Geometry on Anti-De Sitter Space | | Der-Chen Chang
; Irina Markina
; Alexander Vasil'ev
; | | Date: |
7 Aug 2007 | | Abstract: | Sub-Riemannian Geometry is proved to play an important role in many
applications, e.g., Mathematical Physics and Control Theory. The simplest
example of sub-Riemannian structure is provided by the 3-D Heisenberg group.
Sub-Riemannian Geometry enjoys major differences from the Riemannian being a
generalization of the latter at the same time, e.g., geodesics are not unique,
the Hausdorff dimension is larger than the manifold topological dimension.
There exists a large amount of literature developing sub-Riemannian Geometry.
However, very few is known about its natural extension to pseudo-Riemannian
analogues. It is natural to begin such a study with some low-dimensional
manifolds. Based on ideas from sub-Riemannian geometry we develop
sub-Lorentzian geometry over the classical 3-D anti-de Sitter space. Two
different distributions of the tangent bundle of anti-de Sitter space yield two
different geometries: sub-Lorentzian and sub-Riemannian. It is shown that the
set of timelike and spacelike ’horisontal’ curves is non-empty and we study the
problem of horizontal connectivity in anti-de Sitter space. We also use
Lagrangian and Hamiltonian formalisms for both sub-Lorentzian sub-Riemannian
geometries to find geodesics. | | Source: | arXiv, 0708.0879 | | Services: | Forum | Review | PDF | Favorites | |
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