American Institute of Mathematical Sciences

Advanced
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

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.  Google Scholar

[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.  Google Scholar

[3]

, "Sub-Riemannian Geometry", Edited by André Bellaïche and Jean-Jacques Risler, Progress in Mathematics, 144,, Birkhäuser Verlag, (1996).  144,+1996" target="_new" title="Go to article in Google Scholar"> 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), 1851-1878. 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, Cambridge, 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), 721-739. 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), R85-R123. 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), 82-110. 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), 986-1000. 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), 98-105. doi: 10.1007/BF01450011.  Google Scholar

[11]

M. Grochowski, Geodesics in the sub-Lorentzian geometry, Bull. Polish Acad. Sci. Math., 50 (2002), 161-178.  Google Scholar

[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.  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), 145-160. doi: 10.1007/s10450-006-0378-y.  Google Scholar

[14]

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

[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.  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), 104 pp.  Google Scholar

[17]

R. Montgomery, "A Tour of Subriemannian Geometries, Their Geodesics and Applications," Mathematical Surveys and Monographs, 91, American Mathematical Society, Providence, RI, 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), 83-94. Google Scholar

[19]

R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom., 24 (1986), 221-263.  Google Scholar

[20]

R. S. Strichartz, Corrections to: "Sub-Riemannian geometry", J. Differential Geom., 24 (1986), 221-263; J. Differential Geom., 30 (1989), 595-596.  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), 7-17.  Google Scholar

[22]

E. Witten, String theory and black holes, Phys. Rev. D, 44 (1991), 314-324. 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), 321-358. doi: 10.1007/BF02269423.  Google Scholar

[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.  Google Scholar

[3]

, "Sub-Riemannian Geometry", Edited by André Bellaïche and Jean-Jacques Risler, Progress in Mathematics, 144,, Birkhäuser Verlag, (1996).  144,+1996" target="_new" title="Go to article in Google Scholar"> 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), 1851-1878. 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, Cambridge, 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), 721-739. 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), R85-R123. 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), 82-110. 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), 986-1000. 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), 98-105. doi: 10.1007/BF01450011.  Google Scholar

[11]

M. Grochowski, Geodesics in the sub-Lorentzian geometry, Bull. Polish Acad. Sci. Math., 50 (2002), 161-178.  Google Scholar

[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.  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), 145-160. doi: 10.1007/s10450-006-0378-y.  Google Scholar

[14]

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

[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.  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), 104 pp.  Google Scholar

[17]

R. Montgomery, "A Tour of Subriemannian Geometries, Their Geodesics and Applications," Mathematical Surveys and Monographs, 91, American Mathematical Society, Providence, RI, 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), 83-94. Google Scholar

[19]

R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom., 24 (1986), 221-263.  Google Scholar

[20]

R. S. Strichartz, Corrections to: "Sub-Riemannian geometry", J. Differential Geom., 24 (1986), 221-263; J. Differential Geom., 30 (1989), 595-596.  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), 7-17.  Google Scholar

[22]

E. Witten, String theory and black holes, Phys. Rev. D, 44 (1991), 314-324. doi: 10.1103/PhysRevD.44.314.  Google Scholar

[1]

Yunlong Huang, P. S. Krishnaprasad. Sub-Riemannian geometry and finite time thermodynamics Part 1: The stochastic oscillator. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1243-1268. doi: 10.3934/dcdss.2020072
[2]

Paul W. Y. Lee, Chengbo Li, Igor Zelenko. Ricci curvature type lower bounds for sub-Riemannian structures on Sasakian manifolds. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 303-321. doi: 10.3934/dcds.2016.36.303
[3]

Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure & Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307
[4]

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
[5]

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
[6]

Beatrice Abbondanza, Stefano Biagi. Riesz-type representation formulas for subharmonic functions in sub-Riemannian settings. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3161-3192. doi: 10.3934/cpaa.2021101
[7]

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 & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161
[8]

Ali Maalaoui. A note on commutators of the fractional sub-Laplacian on Carnot groups. Communications on Pure & Applied Analysis, 2019, 18 (1) : 435-453. doi: 10.3934/cpaa.2019022
[9]

Jelena Rupčić. Convergence of lacunary SU(1, 1)-valued trigonometric products. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1275-1289. doi: 10.3934/cpaa.2020062
[10]

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
[11]

Fei Liu, Fang Wang, Weisheng Wu. On the Patterson-Sullivan measure for geodesic flows on rank 1 manifolds without focal points. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1517-1554. doi: 10.3934/dcds.2020085
[12]

Fei Liu, Xiaokai Liu, Fang Wang. On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4791-4804. doi: 10.3934/dcds.2021057
[13]

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
[14]

Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Stokes-Brinkman transmission problems on Lipschitz and $C^1$ domains in Riemannian manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 493-537. doi: 10.3934/cpaa.2010.9.493
[15]

Mirela Kohr, Cornel Pintea, Wolfgang L. Wendland. Dirichlet - transmission problems for general Brinkman operators on Lipschitz and $C^1$ domains in Riemannian manifolds. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 999-1018. doi: 10.3934/dcdsb.2011.15.999
[16]

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 & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021249
[17]

Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841
[18]

Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022
[19]

Iacopo P. Longo, Sylvia Novo, Rafael Obaya. Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5491-5520. doi: 10.3934/dcds.2019224
[20]

Sheena D. Branton. Sub-actions for young towers. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 541-556. doi: 10.3934/dcds.2008.22.541

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy