Metrics that realize all Lorentzian holonomy algebras. (English) Zbl 1112.53039

Galaev, Anton S.

Int. J. Geom. Methods Mod. Phys. 3, No. 5-6, 1025-1045 (2006).

Summary: All candidates to the weakly-irreducible not irreducible holonomy algebras of the Lorentzian manifolds are known. In the present paper metrics that realize all these candidates as holonomy algebras are given. This completes the classification of the Lorentzian holonomy algebras. Also new examples of metrics with the holonomy algebras \(g_{2} \ltimes \mathbb R^{7} \subset \mathfrak{so}(1, 8)\) and \(\mathfrak{spin}(7) \ltimes \mathbb R^8 \subset \mathfrak{so}(1, 9)\) are constructed.

MSC:

53C29	Issues of holonomy in differential geometry
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53B30	Local differential geometry of Lorentz metrics, indefinite metrics

Keywords:

Lorentzian manifold; holonomy algebra; local metric

References:

[1]	Alekseevsky D. V., Funksional Anal. i Prilozhen. 2 pp 1–
[2]	Ambrose W., Trans. Amer. Math. Soc. 79 pp 428–
[3]	Berard Bergery L., Bull. Soc. Math. France 125 pp 93–
[4]	DOI: 10.1007/978-3-540-74311-8
[5]	Berger M., Bull. Soc. Math. France 83 pp 279–
[6]	Borel A., C. R. Acad. Sci. Paris 234 pp 279–
[7]	DOI: 10.2307/1971360 · Zbl 0637.53042
[8]	DOI: 10.1007/PL00004860 · Zbl 0997.53051
[9]	DOI: 10.1016/j.difgeo.2004.07.002 · Zbl 1078.53038
[10]	Ikemakhen A., Ann. Sci. Math. Québec 20 pp 53–
[11]	DOI: 10.5565/PUBLMAT_43199_03 · Zbl 0942.53032
[12]	Joyce D., Compact Manifolds with Special Holonomy (2000) · Zbl 1027.53052
[13]	Sfetsos K., J. High Energy Phys. 9 pp 10–
[14]	DOI: 10.2140/pjm.1967.20.351 · Zbl 0149.39603