For citation:
Galaev A. S. Isometry groups of Lobachevskian spaces, similarity transformation groups of Euclidean spaces and Lorentzian holonomy groups. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2005, vol. 5, iss. 1, pp. 3-11. DOI: 10.18500/1816-9791-2005-5-1-3-11, EDN: CYTCBR
This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online:
30.09.2005
Full text:
(downloads: 177)
Language:
Russian
Heading:
UDC:
514.764.214, 512.816.3
EDN:
CYTCBR
Isometry groups of Lobachevskian spaces, similarity transformation groups of Euclidean spaces and Lorentzian holonomy groups
Autors:
Galaev A. S., Saratov State University
Abstract:
In the present paper transitively and simply transitively acting isometry groups of Lobachevskian spaces and transitively acting similarity transformation groups of Euclidean spaces are classified. A geometrical proof of the result of L. Berard Bergery and A. Ikemakhen about the classification of weakly irreducible not irreducible subalgebras of so(1,n+1) is given.
References:
- Borel A., Lichnerowicz A. Groupes d’holonomie des varietes riemanniennes // C. R. Acad. Sci. Paris. 1952. V. 234. P. 279–300.
- Berger M. Sur les groupers d’holonomie des varietes aconnexion affine et des varietes riemanniennes // Bull. Soc. Math. France. 1955. V. 83. P. 279–330.
- Besse A.L. E i n s t e i n m a n i f o l d s . B e r l i n ; H e i d e l b e r g ; N.Y., 1987.
- Joyce D. Compact manifolds with special holonomy. Oxford, 2000.
- Алексеевский Д.В. Римановы пространства с необычными группами голономии // Функциональный анализ и его приложения. 1968. Т. 2, вып. 2. С. 1–10.
- Ambrose W., Singer I.M. A theorem on holonomy // Trans. Amer. Math. Soc. 1953. V. 79. P. 428-443.
- Wu H. Holonomy groups of indefinite metrics // Pacific J. Math. 1967. V. 20. P. 351–382.
- Di Scala A.J., Olmos C. The geometry of homogeneous submanifolds of hyperbolic space // Math. J. 2001. V. 237. P. 199–209.
- Boubel C., Zeghib A. Dynamics of some Lie subgroups of O(n,1) applications // Prepublication de l’ENS Lyon. 2003. № 315.
- Berard Bergery L., Ikemakhen A. On the holonomy of Lorentzian manifolds // Proc. of symp. in pure math. 1993. V. 54. P. 27–40.
- Алексеевский Д.В. Однородные римановы многообразия отрицательной кривизны // Мат. сб. 1975. № 1. С. 93–117.
- Алексеевский Д.В., Винберг Э.Б., Солодовников А.С. Геометрия пространств постоянной кривизны // Итоги науки и техники ВИНИТИ. Совр. пробл. мат. фунд. направления. 1988. Т. 29. С. 5–146.
- Boubel C. On the holonomy of Lorentzian metrics // Prepublication de l’ENS Lyon. 2004. № 323.
- Ikemakhen A. Examples of indecomposable nonirreducible Lorentzian manifolds // Ann. Sci. Math. Quebec. 1996. V. 20, № 1. P. 53–66.
- Leistner T. Berger algebras, weak-Berger algebras and Lorentzian holonomy. Sfb 288-preprint № 567. Berlin, 2002.
- Leistner T. Towards a classification of Lorentzian holonomy groups // ArXiv: math. DG/0305139. 2003.
- Leistner T. Towards a classification of Lorentzian holonomy groups. Part II: semisimple, non-simple weak-Berger algebras // ArXiv: math. DG/0309274. 2003.
- Винберг Э.Б., Онищик А.Л. Семинар по группам Ли и алгебраическим группам. М., 1995.
Received:
17.03.2005
Accepted:
13.08.2005
Published:
30.09.2005
- 907 reads