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

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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

Erlend Grong ^1, and Alexander Vasil’ev ^1,

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

Abstract
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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.

Keywords: SU(1, Carnot-Carathéodory metric, geodesic., sub-Riemannian and sub-Lorentzian manifolds, universal cover, 1), optimal control.

Mathematics Subject Classification: Primary: 53C17, 53B30, 22E30; Secondary: 53C50, 83C6.

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), 321-358. doi: 10.1007/BF02269423.
[2]	A. Agrachev and Yu. Sachkov, "Control Theory From The Geometric Viewpoint," Encyclopaedia of Math. Sci., 87, Control Theory and Optimization, II, Springer-Verlag, Berlin, 2004.
[3]	, "Sub-Riemannian Geometry", Edited by André Bellaïche and Jean-Jacques Risler, Progress in Mathematics, 144,, Birkhäuser Verlag, (1996).
[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), 1851-1878. doi: 10.1137/070703727.
[5]	O. Calin and D.-C. Chang, "Sub-Riemannian Geometry. General Theory and Examples," Encyclopedia of Mathematics and its Applications, 126, Cambridge Univ. Press, Cambridge, 2009.
[6]	O. Calin, D.-C. Chang and I. Markina, Sub-Riemannian geometry of the sphere $S^3$, Canadian J. Math., 61 (2009), 721-739. doi: 10.4153/CJM-2009-039-2.
[7]	S. Carlip, Conformal field theory, $(2+1)$-dimensional gravity and the BTZ black hole, Classical Quantum Gravity, 22 (2005), R85-R123. doi: 10.1088/0264-9381/22/12/R01.
[8]	D.-C. Chang, I. Markina and A. Vasil'ev, Sub-Lorentzian geometry on anti-de Sitter space, J. Math. Pures Appl., 90 (2008), 82-110. doi: 10.1016/j.matpur.2008.02.012.
[9]	D.-C. Chang, I. Markina and A. Vasil'ev, Hopf fibration: geodesics and distances, J. Geom. Phys., 61 (2011), 986-1000. doi: 10.1016/j.geomphys.2011.01.011.
[10]	W. L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117 (1939), 98-105. doi: 10.1007/BF01450011.
[11]	M. Grochowski, Geodesics in the sub-Lorentzian geometry, Bull. Polish Acad. Sci. Math., 50 (2002), 161-178.
[12]	M. Grochowski, On the Heisenberg sub-Lorentzian metric on $\mathbb R^3$, Geometric Singularity Theory, Banach Center Publ., Polish Acad. Sci., Warsaw, 65 (2004), 57-65.
[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), 145-160. doi: 10.1007/s10450-006-0378-y.
[14]	V. Jurdjevic, "Geometric Control Theory," Cambridge Studies in Adv. Math., 52, Cambridge Univ. Press, Cambridge, 1997.
[15]	A. Korolko and I. Markina, Nonholonomic Lorentzian geometry on some $\mathbb H$-type groups, J. Geom. Anal., 19 (2009), 864-889. doi: 10.1007/s12220-009-9088-5.
[16]	W. Liu and H. J. Sussman, Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc., 118 (1995), 104 pp.
[17]	R. Montgomery, "A Tour of Subriemannian Geometries, Their Geodesics and Applications," Mathematical Surveys and Monographs, 91, American Mathematical Society, Providence, RI, 2002.
[18]	P. K. Rashevskiĭ, About connecting two points of complete nonholonomic space by admissible curve, Uch. Zapiski Ped. Inst. K. Liebknecht, 2 (1938), 83-94.
[19]	R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom., 24 (1986), 221-263.
[20]	R. S. Strichartz, Corrections to: "Sub-Riemannian geometry", J. Differential Geom., 24 (1986), 221-263; J. Differential Geom., 30 (1989), 595-596.
[21]	A. M. Vershik and V. Ya. Gershkovich, Geodesic flow on $\SL(2,\mathbb R)$ with nonholonomic restrictions, Zap. Nauchn. Semin. LOMI, 155 (1986), 7-17.
[22]	E. Witten, String theory and black holes, Phys. Rev. D, 44 (1991), 314-324. doi: 10.1103/PhysRevD.44.314.

show all references

References:

[1]	A. Agrachev, Exponential mapping for contact sub-Riemannian structures, Journal of Dynamical and Control Systems, 2 (1996), 321-358. doi: 10.1007/BF02269423.
[2]	A. Agrachev and Yu. Sachkov, "Control Theory From The Geometric Viewpoint," Encyclopaedia of Math. Sci., 87, Control Theory and Optimization, II, Springer-Verlag, Berlin, 2004.
[3]	, "Sub-Riemannian Geometry", Edited by André Bellaïche and Jean-Jacques Risler, Progress in Mathematics, 144,, Birkhäuser Verlag, (1996).
[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), 1851-1878. doi: 10.1137/070703727.
[5]	O. Calin and D.-C. Chang, "Sub-Riemannian Geometry. General Theory and Examples," Encyclopedia of Mathematics and its Applications, 126, Cambridge Univ. Press, Cambridge, 2009.
[6]	O. Calin, D.-C. Chang and I. Markina, Sub-Riemannian geometry of the sphere $S^3$, Canadian J. Math., 61 (2009), 721-739. doi: 10.4153/CJM-2009-039-2.
[7]	S. Carlip, Conformal field theory, $(2+1)$-dimensional gravity and the BTZ black hole, Classical Quantum Gravity, 22 (2005), R85-R123. doi: 10.1088/0264-9381/22/12/R01.
[8]	D.-C. Chang, I. Markina and A. Vasil'ev, Sub-Lorentzian geometry on anti-de Sitter space, J. Math. Pures Appl., 90 (2008), 82-110. doi: 10.1016/j.matpur.2008.02.012.
[9]	D.-C. Chang, I. Markina and A. Vasil'ev, Hopf fibration: geodesics and distances, J. Geom. Phys., 61 (2011), 986-1000. doi: 10.1016/j.geomphys.2011.01.011.
[10]	W. L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann., 117 (1939), 98-105. doi: 10.1007/BF01450011.
[11]	M. Grochowski, Geodesics in the sub-Lorentzian geometry, Bull. Polish Acad. Sci. Math., 50 (2002), 161-178.
[12]	M. Grochowski, On the Heisenberg sub-Lorentzian metric on $\mathbb R^3$, Geometric Singularity Theory, Banach Center Publ., Polish Acad. Sci., Warsaw, 65 (2004), 57-65.
[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), 145-160. doi: 10.1007/s10450-006-0378-y.
[14]	V. Jurdjevic, "Geometric Control Theory," Cambridge Studies in Adv. Math., 52, Cambridge Univ. Press, Cambridge, 1997.
[15]	A. Korolko and I. Markina, Nonholonomic Lorentzian geometry on some $\mathbb H$-type groups, J. Geom. Anal., 19 (2009), 864-889. doi: 10.1007/s12220-009-9088-5.
[16]	W. Liu and H. J. Sussman, Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc., 118 (1995), 104 pp.
[17]	R. Montgomery, "A Tour of Subriemannian Geometries, Their Geodesics and Applications," Mathematical Surveys and Monographs, 91, American Mathematical Society, Providence, RI, 2002.
[18]	P. K. Rashevskiĭ, About connecting two points of complete nonholonomic space by admissible curve, Uch. Zapiski Ped. Inst. K. Liebknecht, 2 (1938), 83-94.
[19]	R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom., 24 (1986), 221-263.
[20]	R. S. Strichartz, Corrections to: "Sub-Riemannian geometry", J. Differential Geom., 24 (1986), 221-263; J. Differential Geom., 30 (1989), 595-596.
[21]	A. M. Vershik and V. Ya. Gershkovich, Geodesic flow on $\SL(2,\mathbb R)$ with nonholonomic restrictions, Zap. Nauchn. Semin. LOMI, 155 (1986), 7-17.
[22]	E. Witten, String theory and black holes, Phys. Rev. D, 44 (1991), 314-324. doi: 10.1103/PhysRevD.44.314.

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