### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 031, 23 pages      arXiv:1312.4581      https://doi.org/10.3842/SIGMA.2015.031

### Invariants and Infinitesimal Transformations for Contact Sub-Lorentzian Structures on 3-Dimensional Manifolds

Marek Grochowski ab and Ben Warhurst c
a) Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszyński University, ul. Dewajtis 5, 01-815 Waszawa, Poland
b) Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-950 Warszawa, Poland
c) Institute of Mathematics, The Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland

Received October 10, 2014, in final form March 30, 2015; Published online April 17, 2015

Abstract
In this article we develop some elementary aspects of a theory of symmetry in sub-Lorentzian geometry. First of all we construct invariants characterizing isometric classes of sub-Lorentzian contact $3$ manifolds. Next we characterize vector fields which generate isometric and conformal symmetries in general sub-Lorentzian manifolds. We then focus attention back to the case where the underlying manifold is a contact $3$ manifold and more specifically when the manifold is also a Lie group and the structure is left-invariant.

Key words: sub-Lorentzian; contact distribution; left-invariant; symmetry.

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