|
Actions of semisimple Lie groups preserving a degenerate Riemannian metric
Author(s):
E.
Bekkara;
C.
Frances;
A.
Zeghib
Journal:
Trans. Amer. Math. Soc.
362
(2010),
2415-2434.
MSC (2010):
Primary 53B30, 53C22, 53C50
Posted:
December 17, 2009
MathSciNet review:
2584605
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
We prove a rigidity of the lightcone in Minkowski space. It is (essentially) the unique space endowed with a lightlike metric and supporting an isometric nonproper action of a semisimple Lie group.
References:
-
- 1.
- S. Adams: Orbit nonproper actions on Lorentz manifolds, Geom. Funct. Anal. 11 (2001), no. 2, 201-243. MR 1837363 (2002f:37048)
- 2.
- S. Adams: Dynamics of semisimple Lie groups on Lorentz manifolds, Geom. Ded. 105 (2004), 1-12. MR 2057240 (2005d:37050)
- 3.
- M. Akivis, V. Goldberg, On some methods of construction of invariant normalizations of lightlike hypersurfaces. Differential Geom. Appl. 12 (2000), no. 2, 121-143. MR 1758845 (2001f:53144)
- 4.
- A. Arouche, M. Deffaf, A. Zeghib , On Lorentz dynamics: From group actions to warped products via homogeneous spaces. Trans. Amer. Math. Soc. 359 (2007), 1253-1263 MR 2262849 (2008b:53091)
- 5.
- E. Bekkara, C. Frances, A. Zeghib, On lightlike geometry: isometric actions, and rigidity aspects. C. R. Math. Acad. Sci. Paris 343 (2006), no. 5, 317-321. MR 2253050 (2007e:53045)
- 6.
- F. Bonsante, Flat spacetimes with compact hyperbolic Cauchy surfaces. J. Differential Geom. 69 (2005), 441-521. MR 2170277 (2006h:53068)
- 7.
- Y. Carrière, Flots riemanniens. Astérisque No. 116 (1984), 31-52. MR 755161 (86m:58125a)
- 8.
- G. D'Ambra, M. Gromov, Lectures on transformation groups: geometry and dynamics. Surveys in differential geometry (Cambridge, MA, 1990), 19-111, Lehigh Univ., Bethlehem, PA, 199. MR 1144526 (93d:58117)
- 9.
- M. Deffaf, K. Melnick, A. Zeghib, Actions of noncompact semisimple groups on Lorentz manifolds, Geom. Funct. Anal. 18 (2008), no. 2, 463-488. MR 2421545 (2009g:57059)
- 10.
- K. Duggal, A. Bejancu, Lightlike submanifolds of semi-Riemannian manifolds and applications. Mathematics and its Applications, 364. Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1383318 (97e:53121)
- 11.
- P Eberlein: Geometry of Nonpositively Curved Manifolds, University of Chicago Press, Chicago, 1996. MR 1441541 (98h:53002)
- 12.
- F. Friedlander, The wave equation on a curved space-time. Cambridge Monographs on Mathematical Physics, No. 2. Cambridge University Press, Cambridge-New York-Melbourne, 1975. MR 0460898 (57:889)
- 13.
- S. Hawking, G. Ellis, The large scale structure of space-time. Cambridge Monographs on Mathematical Physics, No. 1. Cambridge University Press, London-New York, 1973. MR 0424186 (54:12154)
- 14.
- A. Knapp, Lie groups beyond an introduction. Second edition. Progress in Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389 (2003c:22001)
- 15.
- T. Kobayashi, T. Yoshino, Compact Clifford-Klein form of symmetric spaces -revisited, Pure Appl. Math. Q. 1, (2005), no. 3, 591-663. MR 2201328 (2007h:22013)
- 16.
- N. Kowalsky, Noncompact simple automorphism groups of Lorentz manifolds. Ann. Math. 144 (1997), 611-640. MR 1426887 (98g:57059)
- 17.
- D. Kupeli, Singular semi-Riemannian geometry. With the collaboration of Eduardo Garca-Ro on Part III. Mathematics and its Applications, 366. Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1392222 (97f:53105)
- 18.
- T. Miernowski, Thesis, école normale supérieure de Lyon, 2005.
- 19.
- P. Molino, Riemannian foliations. Progress in Mathematics, 73. Birkhuser Boston, Inc., Boston, MA, 1988. MR 932463 (89b:53054)
- 20.
- P. Chruściel, On rigidity of analytic black holes. Comm. Math. Phys. 189 (1997), 1-7. MR 1478527 (98j:83054)
- 21.
- I. Robinson, A. Trautman, Integrable optical geometry, Lett. Math. Phys. 10 (1985), 179-182. MR 815241 (87f:53070)
- 22.
- K. Scannell, Flat conformal structures and the classification of de Sitter manifolds, Comm. Anal. Geom. 7 (1999), no. 2, 325-345. MR 1685590 (2000a:53122)
- 23.
- A. Zeghib, Geodesic foliations in Lorentz
 -manifolds. Comment. Math. Helv. 74 (1999), 1-21. MR 1677118 (99k:53049) - 24.
- A. Zeghib, Isometry groups and geodesic foliations of Lorentz manifolds. II. Geometry of analytic Lorentz manifolds with large isometry groups. Geom. Funct. Anal. 9 (1999), 823-854. MR 1719610 (2001g:53060)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical
Society
with
MSC (2010):
53B30, 53C22, 53C50
Retrieve articles in all Journals with
MSC (2010):
53B30, 53C22, 53C50
Additional Information:
E.
Bekkara
Affiliation:
Department of Mathematics and Information, BP1523, ENSET of Oran, 31000 El M'naouar Oran, Algeria
Email:
esmaa.bekkara@gmail.com
C.
Frances
Affiliation:
Laboratoire de Mathématiques, Université Paris-Sud, 91405 Orsay Cedex, France
Email:
charles.frances@math.u-psud.fr
A.
Zeghib
Affiliation:
CNRS, UMPA, École Normale Supérieure de Lyon, 69364 Lyon Cexex 07, France
Email:
zeghib@umpa.ens-lyon.fr
DOI:
10.1090/S0002-9947-09-05030-2
PII:
S 0002-9947(09)05030-2
Keywords:
Lightlike metric,
lightcone,
isotropic direction
Received by editor(s):
November 26, 2007
Posted:
December 17, 2009
Additional Notes:
The first author was partially supported by the project CMEP 05 MDU 641B of the Tassili program.
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
