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), 321-358. 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.

[1]

Yunlong Huang, P. S. Krishnaprasad. Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1243-1268. doi: 10.3934/dcdss.2020072

[2]

Lucas Dahinden, Álvaro del Pino. Introducing sub-Riemannian and sub-Finsler billiards. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3187-3232. doi: 10.3934/dcds.2022014

[3]

Paul W. Y. Lee, Chengbo Li, Igor Zelenko. Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 303-321. doi: 10.3934/dcds.2016.36.303

[4]

Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure and Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307

[5]

Daniel Genin, Serge Tabachnikov. On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards. Journal of Modern Dynamics, 2007, 1 (2) : 155-173. doi: 10.3934/jmd.2007.1.155

[6]

Stefan Sommer, Anne Marie Svane. Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry. Journal of Geometric Mechanics, 2017, 9 (3) : 391-410. doi: 10.3934/jgm.2017015

[7]

Beatrice Abbondanza, Stefano Biagi. Riesz-type representation formulas for subharmonic functions in sub-Riemannian settings. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3161-3192. doi: 10.3934/cpaa.2021101

[8]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[9]

Ali Maalaoui. A note on commutators of the fractional sub-Laplacian on Carnot groups. Communications on Pure and Applied Analysis, 2019, 18 (1) : 435-453. doi: 10.3934/cpaa.2019022

[10]

Jelena Rupčić. Convergence of lacunary SU(1, 1)-valued trigonometric products. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1275-1289. doi: 10.3934/cpaa.2020062

[11]

Anthony M. Bloch, Rohit Gupta, Ilya V. Kolmanovsky. Neighboring extremal optimal control for mechanical systems on Riemannian manifolds. Journal of Geometric Mechanics, 2016, 8 (3) : 257-272. doi: 10.3934/jgm.2016007

[12]

Fei Liu, Fang Wang, Weisheng Wu. On the Patterson-Sullivan measure for geodesic flows on rank 1 manifolds without focal points. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1517-1554. doi: 10.3934/dcds.2020085

[13]

Fei Liu, Xiaokai Liu, Fang Wang. On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4791-4804. doi: 10.3934/dcds.2021057

[14]

Antonella Marini, Thomas H. Otway. Strong solutions to a class of boundary value problems on a mixed Riemannian--Lorentzian metric. Conference Publications, 2015, 2015 (special) : 801-808. doi: 10.3934/proc.2015.0801

[15]

Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Stokes-Brinkman transmission problems on Lipschitz and $C^1$ domains in Riemannian manifolds. Communications on Pure and Applied Analysis, 2010, 9 (2) : 493-537. doi: 10.3934/cpaa.2010.9.493

[16]

Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Dirichlet - transmission problems for general Brinkman operators on Lipschitz and $C^1$ domains in Riemannian manifolds. Discrete and Continuous Dynamical Systems - B, 2011, 15 (4) : 999-1018. doi: 10.3934/dcdsb.2011.15.999

[17]

Ta Cong Son, Nguyen Tien Dung, Nguyen Van Tan, Tran Manh Cuong, Hoang Thi Phuong Thao, Pham Dinh Tung. Weak convergence of delay SDEs with applications to Carathéodory approximation. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021249

[18]

Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841

[19]

Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022

[20]

Iacopo P. Longo, Sylvia Novo, Rafael Obaya. Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5491-5520. doi: 10.3934/dcds.2019224

2020 Impact Factor: 0.857

Metrics

  • PDF downloads (98)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]