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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 |
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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