On the holonomy of Lorentzian metrics
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 3, p. 427-475

Indecomposable Lorentzian holonomy algebras, except 𝔰𝔬(n,1) and {0}, are not semi-simple; they possibly belong to four families of algebras. All four families are realized as families of holonomy algebras: we describe the corresponding set of germs of metrics in each case.

Les algèbres d’holonomie lorentziennes indécomposables, exceptées 𝔰𝔬(n,1) et {0}, ne sont pas semi-simples. Elles se classent en quatre familles possibles. Ces quatre familles sont effectivement réalisées commes familles d’algèbres d’holonomie  : nous décrivons, pour chacune d’elles, l’ensemble correspondant de germes de métriques.

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     author = {Boubel, Charles},
     title = {On the holonomy of Lorentzian metrics},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 16},
     number = {3},
     year = {2007},
     pages = {427-475},
     doi = {10.5802/afst.1156},
     mrnumber = {2379049},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2007_6_16_3_427_0}
}
Boubel, Charles. On the holonomy of Lorentzian metrics. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 3, pp. 427-475. doi : 10.5802/afst.1156. http://www.numdam.org/item/AFST_2007_6_16_3_427_0/

[BBI93] Bérard Bergery (L.), Ikemakhen (A.).— On the Holonomy of Lorentzian Manifolds, Proc. of Symposia in Pure Mathematics 54, Part 2, p. 27-39 (1993). | MR 1216527 | Zbl 0807.53014

[Be55] Berger (M.).— Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes. (French) Bull. Soc. Math. France 83, p. 279-330 (1955). | Numdam | Zbl 0068.36002

[Be57] Berger (M.).— Les espaces symétriques non compacts, Ann. Sci. Ecol. Norm. Sup. 74, p. 85-177 (1957). | Numdam | MR 104763 | Zbl 0093.35602

[Bes87] Besse (A.L.).— Einstein Manifolds. Springer Verlag — Berlin, Heidelberg (1987). | MR 867684 | Zbl 0613.53001

[Br96] Bryant (R.).— Classical, exceptional, and exotic holonomies: A status report, Besse, A. L. (ed.), Actes de la table ronde de géométrie différentielle en l’honneur de Marcel Berger, Luminy, France, 12–18 juillet 1992. Soc. Math. France. Sémin. Congr. 1, p. 93-165 (1996). | Zbl 0882.53014

[Bo00] Boubel (C.).— Sur l’holonomie des variétés pseudo-riemanniennes, Thèse de doctorat, Université Nancy I, France, May 2000.

[BZ03] Boubel (C.), Zeghib (A.).— Isometric actions of Lie subgroups of the Moebius group, Nonlinearity, 17, p. 1677-1688 (2004). | MR 2086144 | Zbl 1073.53016

[CP80] Cahen (M.), Parker (M.).— Pseudo-Riemannian symmetric spaces, Memoirs of the American Mathematical Society, 24 (229) (1980). | MR 556610 | Zbl 0438.53057

[C24] Cartan (E.).— Sur les variétés à connexion affine et la théorie de la relativité généralisée I et II, Ann. Sci. Ecol. Norm. Sup. 40, p. 325-412 (1923) and 41, p. 1-25 (1924). | JFM 49.0542.02

[C26] Cartan (E.).— Les groupes d’holonomie des espaces généralisés, Acta. Math. 48, p. 1-42 (1926).

[E68] Ebin. (D.G.).— The manifold of Riemannian metrics, 1970 Global Analysis, Proc. of Symposia in Pure Mathematics 15, p. 11-40 (1968). | MR 267604 | Zbl 0205.53702

[G05] Galaev (A.).— Metrics that realize all types of Lorentzian holonomy, preprint, http://arxiv.org/abs/math.DG/0502575.

[HO56] Hano (J.), Ozeki (H.).— On the holonomy group of linear connections, Nagoya Math. J. 10, p. 97-100 (1956). | MR 82164 | Zbl 0070.38901

[I96] Ikemakhen (A.).— Examples of indecomposable non-irreducible Lorentzian manifolds. Ann. Sci. Math. Qué. 20, No.1, p. 53-66 (1996). | MR 1397338 | Zbl 0873.53009

[LB70] Lascoux (A.), Berger (M.).— Variétés kählériennes compactes. (French) Lecture Notes in Mathematics. 154, Berlin-Heidelberg-New York: Springer-Verlag (1970). | MR 278248 | Zbl 0205.51702

[L03a] Leistner (T.).— Towards a classification of Lorentzian holonomy groups, Preprint, 2003, http://arxiv.org/ps/math.DG/0305139 and Part II: Semisimple, non-simple weak-Berger algebras. Preprint, 2003, http://arxiv.org/ps/math.DG/0309274.

[L03b] Leistner (T.).— Holonomy and Parallel Spinors in Lorentzian Geometry, thesis, Humboldt-Universität zu Berlin (2003). Published by Logos Verlag, Berlin, 2004. | Zbl 1088.53032

[dR52] de Rham (G.).— Sur la réductibilité d’un espace de Riemann, Comm. Math. Helv. 26, p. 328-344 (1952). | Zbl 0048.15701

[DO01] Di Scala (A.J.), Olmos (C.).— The geometry of homogeneous submanifolds of hyperbolic space, Math. Z. 237, No.1, p. 199-209 (2001). | MR 1836778 | Zbl 0997.53051

[S01a] Schwachhöfer (L.).— Irreducible holonomy representations, Proceedings of the 20th Winter School “Geometry and Physics” (Srní, 2000). Rend. Circ. Mat. Palermo (2) Suppl. No. 66, p. 59-81 (2001). | Zbl 1037.53034

[S01b] Schwachhöfer (L.).— Connections with irreducible holonomy representations, Adv. Math. 160, no.1, p. 1-80 (2001). | MR 1831947 | Zbl 1037.53035

[W49] Walker (A.G.).— On parallel fields of partially null vector spaces, Q. J. Math., Oxf. II. Ser. 1, p. 69-79 (sept. 1949). | MR 35085 | Zbl 0036.38303

[W50] Walker (A.G.).— Canonical form for a riemannian space with a parallel field of null planes, I and II, Q. J. Math., Oxf. II. Ser. 1, p. 69-79 and p. 147-152 (1950). | MR 35085 | Zbl 0036.38303

[W67] Wu (H.).— Holonomy groups of indefinite metrics, Pacific J. Math. 20, p. 351-392 (1967). | MR 212740 | Zbl 0149.39603

[Z02] Zeghib (A.).— Remarks on Lorentz symmetric spaces, Compositio Math., 140, no. 6, p. 1675-1678 (2004). | MR 2098408 | Zbl 1067.53009