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
References:
[1]

A. Agrachev, Exponential mapping for contact sub-Riemannian structures,, Journal of Dynamical and Control Systems, 2 (1996), 321.  doi: 10.1007/BF02269423.  Google Scholar

[2]

A. Agrachev and Yu. Sachkov, "Control Theory From The Geometric Viewpoint,", Encyclopaedia of Math. Sci., 87 (2004).   Google Scholar

[3]

, "Sub-Riemannian Geometry", Edited by André Bellaïche and Jean-Jacques Risler, Progress in Mathematics, 144,, Birkhäuser Verlag, (1996).   Google Scholar

[4]

U. Boscain and F. Rossi, Invariant Carnot-Caratheodory metric on $S^3$, $SO(3)$, $SL(2)$ and lens spaces,, SIAM J. Control Optim., 47 (2008), 1851.  doi: 10.1137/070703727.  Google Scholar

[5]

O. Calin and D.-C. Chang, "Sub-Riemannian Geometry. General Theory and Examples," Encyclopedia of Mathematics and its Applications, 126,, Cambridge Univ. Press, (2009).   Google Scholar

[6]

O. Calin, D.-C. Chang and I. Markina, Sub-Riemannian geometry of the sphere $S^3$,, Canadian J. Math., 61 (2009), 721.  doi: 10.4153/CJM-2009-039-2.  Google Scholar

[7]

S. Carlip, Conformal field theory, $(2+1)$-dimensional gravity and the BTZ black hole,, Classical Quantum Gravity, 22 (2005).  doi: 10.1088/0264-9381/22/12/R01.  Google Scholar

[8]

D.-C. Chang, I. Markina and A. Vasil'ev, Sub-Lorentzian geometry on anti-de Sitter space,, J. Math. Pures Appl., 90 (2008), 82.  doi: 10.1016/j.matpur.2008.02.012.  Google Scholar

[9]

D.-C. Chang, I. Markina and A. Vasil'ev, Hopf fibration: geodesics and distances,, J. Geom. Phys., 61 (2011), 986.  doi: 10.1016/j.geomphys.2011.01.011.  Google Scholar

[10]

W. L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung,, Math. Ann., 117 (1939), 98.  doi: 10.1007/BF01450011.  Google Scholar

[11]

M. Grochowski, Geodesics in the sub-Lorentzian geometry,, Bull. Polish Acad. Sci. Math., 50 (2002), 161.   Google Scholar

[12]

M. Grochowski, On the Heisenberg sub-Lorentzian metric on $\mathbb R^3$,, Geometric Singularity Theory, 65 (2004), 57.   Google Scholar

[13]

M. Grochowski, Reachable sets for the Heisenberg sub-Lorentzian structure $\mathbb R^3$. An estimate for the distance function,, J. Dynamical and Control Systems, 12 (2006), 145.  doi: 10.1007/s10450-006-0378-y.  Google Scholar

[14]

V. Jurdjevic, "Geometric Control Theory,", Cambridge Studies in Adv. Math., 52 (1997).   Google Scholar

[15]

A. Korolko and I. Markina, Nonholonomic Lorentzian geometry on some $\mathbb H$-type groups,, J. Geom. Anal., 19 (2009), 864.  doi: 10.1007/s12220-009-9088-5.  Google Scholar

[16]

W. Liu and H. J. Sussman, Shortest paths for sub-Riemannian metrics on rank-two distributions,, Mem. Amer. Math. Soc., 118 (1995).   Google Scholar

[17]

R. Montgomery, "A Tour of Subriemannian Geometries, Their Geodesics and Applications,", Mathematical Surveys and Monographs, 91 (2002).   Google Scholar

[18]

P. K. Rashevskiĭ, About connecting two points of complete nonholonomic space by admissible curve,, Uch. Zapiski Ped. Inst. K. Liebknecht, 2 (1938), 83.   Google Scholar

[19]

R. S. Strichartz, Sub-Riemannian geometry,, J. Differential Geom., 24 (1986), 221.   Google Scholar

[20]

R. S. Strichartz, Corrections to: "Sub-Riemannian geometry", J. Differential Geom., 24 (1986), 221-263;, J. Differential Geom., 30 (1989), 595.   Google Scholar

[21]

A. M. Vershik and V. Ya. Gershkovich, Geodesic flow on $\SL(2,\mathbb R)$ with nonholonomic restrictions,, Zap. Nauchn. Semin. LOMI, 155 (1986), 7.   Google Scholar

[22]

E. Witten, String theory and black holes,, Phys. Rev. D, 44 (1991), 314.  doi: 10.1103/PhysRevD.44.314.  Google Scholar

show all references

References:
[1]

A. Agrachev, Exponential mapping for contact sub-Riemannian structures,, Journal of Dynamical and Control Systems, 2 (1996), 321.  doi: 10.1007/BF02269423.  Google Scholar

[2]

A. Agrachev and Yu. Sachkov, "Control Theory From The Geometric Viewpoint,", Encyclopaedia of Math. Sci., 87 (2004).   Google Scholar

[3]

, "Sub-Riemannian Geometry", Edited by André Bellaïche and Jean-Jacques Risler, Progress in Mathematics, 144,, Birkhäuser Verlag, (1996).   Google Scholar

[4]

U. Boscain and F. Rossi, Invariant Carnot-Caratheodory metric on $S^3$, $SO(3)$, $SL(2)$ and lens spaces,, SIAM J. Control Optim., 47 (2008), 1851.  doi: 10.1137/070703727.  Google Scholar

[5]

O. Calin and D.-C. Chang, "Sub-Riemannian Geometry. General Theory and Examples," Encyclopedia of Mathematics and its Applications, 126,, Cambridge Univ. Press, (2009).   Google Scholar

[6]

O. Calin, D.-C. Chang and I. Markina, Sub-Riemannian geometry of the sphere $S^3$,, Canadian J. Math., 61 (2009), 721.  doi: 10.4153/CJM-2009-039-2.  Google Scholar

[7]

S. Carlip, Conformal field theory, $(2+1)$-dimensional gravity and the BTZ black hole,, Classical Quantum Gravity, 22 (2005).  doi: 10.1088/0264-9381/22/12/R01.  Google Scholar

[8]

D.-C. Chang, I. Markina and A. Vasil'ev, Sub-Lorentzian geometry on anti-de Sitter space,, J. Math. Pures Appl., 90 (2008), 82.  doi: 10.1016/j.matpur.2008.02.012.  Google Scholar

[9]

D.-C. Chang, I. Markina and A. Vasil'ev, Hopf fibration: geodesics and distances,, J. Geom. Phys., 61 (2011), 986.  doi: 10.1016/j.geomphys.2011.01.011.  Google Scholar

[10]

W. L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung,, Math. Ann., 117 (1939), 98.  doi: 10.1007/BF01450011.  Google Scholar

[11]

M. Grochowski, Geodesics in the sub-Lorentzian geometry,, Bull. Polish Acad. Sci. Math., 50 (2002), 161.   Google Scholar

[12]

M. Grochowski, On the Heisenberg sub-Lorentzian metric on $\mathbb R^3$,, Geometric Singularity Theory, 65 (2004), 57.   Google Scholar

[13]

M. Grochowski, Reachable sets for the Heisenberg sub-Lorentzian structure $\mathbb R^3$. An estimate for the distance function,, J. Dynamical and Control Systems, 12 (2006), 145.  doi: 10.1007/s10450-006-0378-y.  Google Scholar

[14]

V. Jurdjevic, "Geometric Control Theory,", Cambridge Studies in Adv. Math., 52 (1997).   Google Scholar

[15]

A. Korolko and I. Markina, Nonholonomic Lorentzian geometry on some $\mathbb H$-type groups,, J. Geom. Anal., 19 (2009), 864.  doi: 10.1007/s12220-009-9088-5.  Google Scholar

[16]

W. Liu and H. J. Sussman, Shortest paths for sub-Riemannian metrics on rank-two distributions,, Mem. Amer. Math. Soc., 118 (1995).   Google Scholar

[17]

R. Montgomery, "A Tour of Subriemannian Geometries, Their Geodesics and Applications,", Mathematical Surveys and Monographs, 91 (2002).   Google Scholar

[18]

P. K. Rashevskiĭ, About connecting two points of complete nonholonomic space by admissible curve,, Uch. Zapiski Ped. Inst. K. Liebknecht, 2 (1938), 83.   Google Scholar

[19]

R. S. Strichartz, Sub-Riemannian geometry,, J. Differential Geom., 24 (1986), 221.   Google Scholar

[20]

R. S. Strichartz, Corrections to: "Sub-Riemannian geometry", J. Differential Geom., 24 (1986), 221-263;, J. Differential Geom., 30 (1989), 595.   Google Scholar

[21]

A. M. Vershik and V. Ya. Gershkovich, Geodesic flow on $\SL(2,\mathbb R)$ with nonholonomic restrictions,, Zap. Nauchn. Semin. LOMI, 155 (1986), 7.   Google Scholar

[22]

E. Witten, String theory and black holes,, Phys. Rev. D, 44 (1991), 314.  doi: 10.1103/PhysRevD.44.314.  Google Scholar

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