June  2011, 3(2): 225-260. doi: 10.3934/jgm.2011.3.225

Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover

1. 

Department of Mathematics, University of Bergen, P.O. Box 7803, Bergen N-5020, Norway, Norway

Received  July 2010 Revised  June 2011 Published  July 2011

We study sub-Riemannian and sub-Lorentzian geometry on the Lie group $SU(1,1)$ and on its universal cover SU~(1,1). In the sub-Riemannian case we find the distance function and completely describe sub-Riemannian geodesics on both $SU(1,1)$ and SU~(1,1), connecting two fixed points. In particular, we prove that there is a strong connection between the conjugate loci and the number of geodesics. In the sub-Lorentzian case, we describe the geodesics connecting two points on SU~(1,1), and compare them with Lorentzian ones. It turns out that the reachable sets for Lorentzian and sub-Lorentzian normal geodesics intersect but are not included one to the other. A description of the timelike future is obtained and compared in the Lorentzian and sub-Lorentzain cases.
Citation: Erlend Grong, Alexander Vasil’ev. Sub-Riemannian and sub-Lorentzian geometry on $SU(1,1)$ and on its universal cover. Journal of Geometric Mechanics, 2011, 3 (2) : 225-260. doi: 10.3934/jgm.2011.3.225
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show all references

References:
[1]

Journal of Dynamical and Control Systems, 2 (1996), 321-358. doi: 10.1007/BF02269423.  Google Scholar

[2]

Encyclopaedia of Math. Sci., 87, Control Theory and Optimization, II, Springer-Verlag, Berlin, 2004.  Google Scholar

[3]

Birkhäuser Verlag, Basel, 1996.  Google Scholar

[4]

SIAM J. Control Optim., 47 (2008), 1851-1878. doi: 10.1137/070703727.  Google Scholar

[5]

Cambridge Univ. Press, Cambridge, 2009.  Google Scholar

[6]

Canadian J. Math., 61 (2009), 721-739. doi: 10.4153/CJM-2009-039-2.  Google Scholar

[7]

Classical Quantum Gravity, 22 (2005), R85-R123. doi: 10.1088/0264-9381/22/12/R01.  Google Scholar

[8]

J. Math. Pures Appl., 90 (2008), 82-110. doi: 10.1016/j.matpur.2008.02.012.  Google Scholar

[9]

J. Geom. Phys., 61 (2011), 986-1000. doi: 10.1016/j.geomphys.2011.01.011.  Google Scholar

[10]

Math. Ann., 117 (1939), 98-105. doi: 10.1007/BF01450011.  Google Scholar

[11]

Bull. Polish Acad. Sci. Math., 50 (2002), 161-178.  Google Scholar

[12]

Geometric Singularity Theory, Banach Center Publ., Polish Acad. Sci., Warsaw, 65 (2004), 57-65.  Google Scholar

[13]

J. Dynamical and Control Systems, 12 (2006), 145-160. doi: 10.1007/s10450-006-0378-y.  Google Scholar

[14]

Cambridge Studies in Adv. Math., 52, Cambridge Univ. Press, Cambridge, 1997.  Google Scholar

[15]

J. Geom. Anal., 19 (2009), 864-889. doi: 10.1007/s12220-009-9088-5.  Google Scholar

[16]

Mem. Amer. Math. Soc., 118 (1995), 104 pp.  Google Scholar

[17]

Mathematical Surveys and Monographs, 91, American Mathematical Society, Providence, RI, 2002.  Google Scholar

[18]

Uch. Zapiski Ped. Inst. K. Liebknecht, 2 (1938), 83-94. Google Scholar

[19]

J. Differential Geom., 24 (1986), 221-263.  Google Scholar

[20]

J. Differential Geom., 30 (1989), 595-596.  Google Scholar

[21]

Zap. Nauchn. Semin. LOMI, 155 (1986), 7-17.  Google Scholar

[22]

Phys. Rev. D, 44 (1991), 314-324. doi: 10.1103/PhysRevD.44.314.  Google Scholar

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