Holonomy groups of 𝐺₂*-manifolds

Fino, Anna; Kath, Ines

doi:10.1090/tran/7427

Holonomy groups of ${\textrm {G}_{2}^*}$-manifolds
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by Anna Fino and Ines Kath PDF

Trans. Amer. Math. Soc. 371 (2019), 7725-7755 Request permission

Abstract:

We classify the holonomy algebras of manifolds admitting an indecomposable torsion-free ${\textrm {G}_{2}^*}$-structure, i.e., for which the holonomy representation does not leave invariant any proper nondegenerate subspace. We realise some of these Lie algebras as holonomy algebras of left-invariant metrics on Lie groups.

References

I, M. Anderson, T. Leistner, and P. Nurowski, Explicit ambient metrics and holonomy, arXiv:1501.00852.
W. Ambrose and I. M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), 428–443. MR 63739, DOI 10.1090/S0002-9947-1953-0063739-1
Marcel Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955), 279–330 (French). MR 79806
Natalia I. Bezvitnaya, Holonomy groups of pseudo-quaternionic-Kählerian manifolds of non-zero scalar curvature, Ann. Global Anal. Geom. 39 (2011), no. 1, 99–105. MR 2745506, DOI 10.1007/s10455-010-9234-0
Robert L. Bryant, Classical, exceptional, and exotic holonomies: a status report, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 93–165 (English, with English and French summaries). MR 1427757
A. Fino and I. Kath, Local metrics with holonomy contained in ${\textrm {G}_{2}^*}$, arXiv:1705.00023.
Anna Fino and Ignacio Luján, Torsion-free $G^*_{2(2)}$-structures with full holonomy on nilmanifolds, Adv. Geom. 15 (2015), no. 3, 381–392. MR 3365752, DOI 10.1515/advgeom-2015-0015
M. Freibert, Calibrated and parallel structures on almost Abelian Lie algebras, arXiv:1307.2542.
A. S. Galaev, Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2, arXiv:math/0612392.
A. S. Galaev, Holonomy groups of Lorentzian manifolds, Uspekhi Mat. Nauk 70 (2015), no. 2(422), 55–108 (Russian, with Russian summary); English transl., Russian Math. Surveys 70 (2015), no. 2, 249–298. MR 3353127, DOI 10.4213/rm9610
Anton Galaev and Thomas Leistner, Recent developments in pseudo-Riemannian holonomy theory, Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lect. Math. Theor. Phys., vol. 16, Eur. Math. Soc., Zürich, 2010, pp. 581–627. MR 2681602, DOI 10.4171/079-1/17
C. Robin Graham and Travis Willse, Parallel tractor extension and ambient metrics of holonomy split $G_2$, J. Differential Geom. 92 (2012), no. 3, 463–505. MR 3005060
I. Kath, $G_{2(2)}^*$-structures on pseudo-Riemannian manifolds, J. Geom. Phys. 27 (1998), no. 3-4, 155–177. MR 1645016, DOI 10.1016/S0393-0440(97)00073-9
Ines Kath, Indefinite symmetric spaces with $\rm G_{2(2)}$-structure, J. Lond. Math. Soc. (2) 87 (2013), no. 3, 853–876. MR 3073680, DOI 10.1112/jlms/jds068
Thomas Leistner, On the classification of Lorentzian holonomy groups, J. Differential Geom. 76 (2007), no. 3, 423–484. MR 2331527
Thomas Leistner and PawełNurowski, Ambient metrics with exceptional holonomy, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 2, 407–436. MR 3011997
Travis Willse, Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy, Differential Geom. Appl. 33 (2014), no. suppl., 81–111. MR 3159952, DOI 10.1016/j.difgeo.2013.10.010