Details

Reza Mirzaie ^[a]

Riemannian \(G\)-manifolds of constant negative curvature whose all orbits are principal

Pages: 45-51
Received: 24 February 2018
Accepted in revised form: 7 August 2018
Mathematics Subject Classification (2010): 53C30, 57S25.
Keywords: Riemannian manifold, Lie group, Isometry.
Author address:
[a]: Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University (IKIU), Qazvin, Iran

Abstract: We give a topological classification on Riemannian \(G\)-manifolds of constant negative curvature and their orbits, under the condition that all orbits are principal.

References

[1]: J. Berndt and M. Brück, Cohomogeneity one actions on hyperbolic spaces, J. Reine Angew. Math. 541 (2001), 209-235. MR1876290
[2]: J. Berndt and H. Tamaru, Homogeneous codimension one foliations on noncompact symmetric spaces, J. Differential Geom. 63 (2003), 1-40. MR2015258
[3]: R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49. MR0251664
[4]: B. H. Bowditch, Discrete parabolic groups, J. Differential Geom. 38 (1993), 559-583. MR1243787
[5]: G. E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, 46, Academic Press, New York-London, 1972. MR0413144
[6]: W. Byers, Isometry groups of manifolds of negative curvature, Proc. Amer. Math. Soc. 54 (1976), 281-285. MR0390960
[7]: J. Dadok, Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288 (1985), 125-137. MR0773051
[8]: M. P. Do Carmo, Riemannian geometry, Math. Theory Appl., Birkhäuser, Boston, MA, 1992. MR1138207
[9]: P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45-109. MR0336648
[10]: P. Eberlein, Geodesic flows in manifolds of nonpositive curvature, http://www.unc.edu/math/Faculty/pbe/AMS_Summer.pdf
[11]: E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23-34. MR0353210
[12]: S. Kobayashi, Homogeneous Riemannian manifolds of negative curvature, Tôhoku Math. J. (2) 14 (1962), 413-415. MR0148015
[13]: P. W. Michor, Isometric actions of Lie groups and invariants, Lecture course at the University of Vienna 1996/97.
[14]: R. Mirzaie, On negatively curved \(G\)-manifolds of low cohomogeneity, Hokkaido Math. J. 38 (2009), 797-803. MR2561960
[15]: R. Mirzaie, On Riemannian manifolds of constant negative curvature, J. Korean Math. Soc. 48 (2011), 23-21. MR2778023
[16]: R. Mirzaie, Actions without nontrivial singular orbits on Riemannian manifolds of negative curvature, Acta Math. Hungar. 147 (2015), 172-178. MR3391520
[17]: R. Mirzaie and S. M. B. Kashani, On cohomogeneity one flat Riemannian manifolds, Glasg. Math. J. 44 (2002), 185-190. MR1902396
[18]: R. Mirzaie, On orbits of isometric actions on flat Riemannian manifolds, Kyushu J. Math. 65 (2011), 383-393. MR2977766
[19]: B. O'Neill, Semi-Riemannian geometry, With applications to relativity, Pure Appl. Math., 103, Academic Press, New York, 1983. MR0719023
[20]: R. S. Palais and C.-L. Terng, A general theory of canonical forms, Trans. Amer. Math. Soc. 300 (1987), 771-789. MR0876478
[21]: F. Podestà and A. Spiro, Some topological properties of cohomogeneity one manifolds with negative curvature, Ann. Global Anal. Geom. 14 (1996), 69-79. MR1375067
[22]: A. J. Di Scala and C. Olmos, The geometry of homogeneous submanifolds of hyperbolic space, Math. Z. 237 (2001), 199-209. MR1836778
[23]: A. J. Di Scala and C. Olmos, A geometric proof of the Karpelevich-Mostow theorem, Bull. Lond. Math. Soc. 41 (2009), 634-638. MR2521358
[24]: J. A. Wolf, Homogeneity and bounded isometries in manifolds of negative curvature, Illinois J. Math. 8 (1964), 14-18. MR0163262