Holonomy groups of Lorentzian manifolds

A. S. Galaev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 70 (2015) no. 2, pp. 249-298

Résumé

This paper contains a survey of recent results on classification of the connected holonomy groups of Lorentzian manifolds. A simplification of the construction of Lorentzian metrics with all possible connected holonomy groups is obtained. The Einstein equation, Lorentzian manifolds with parallel and recurrent spinor fields, conformally flat Walker metrics, and the classification of 2-symmetric Lorentzian manifolds are considered as applications. Bibliography: 123 titles.

Détail

Keywords: Lorentzian manifold, holonomy group, holonomy algebra, Walker manifold, Einstein equation, recurrent spinor field, conformally flat manifold, 2-symmetric Lorentzian manifold.

Bibliographie

[1] V. P. Akulov, D. V. Volkov, V. A. Soroka, “Gauge fields on superspaces with different holonomy groups”, JETP Lett., 22:7 (1975), 187–188

[2] D. V. Alekseevskii, “Riemannian spaces with exceptional holonomy groups”, Funct. Anal. Appl., 2:2 (1968), 97–105 | DOI | MR | Zbl

[3] D. V. Alekseevskii, “Homogeneous Riemannian spaces of negative curvature”, Math. USSR-Sb., 25:1 (1975), 87–109 | DOI | MR | Zbl

[4] D. V. Alekseevsky, V. Cortés, A. S. Galaev, T. Leistner, “Cones over pseudo-Riemannian manifolds and their holonomy”, J. Reine Angew. Math., 635 (2009), 23–69 | DOI | MR | Zbl

[5] D. V. Alekseevsky, A. S. Galaev, “Two-symmetric Lorentzian manifolds”, J. Geom. Phys., 61:12 (2011), 2331–2340 | DOI | MR | Zbl

[6] D. V. Alekseevskii, B. N. Kimel'fel'd, “Classification of homogeneous conformally flat Riemannian manifolds”, Math. Notes, 24:1 (1978), 559–562 | DOI | MR | Zbl

[7] D. V. Alekseevskij, È. B. Vinberg, A. S. Solodovnikov, “Geometry of spaces of constant curvature”, Geometry, II, Encyclopaedia Math. Sci., 29, Springer, Berlin, 1993, 1–138 | DOI | MR | Zbl

[8] W. Ambrose, I. M. Singer, “A theorem on holonomy”, Trans. Amer. Math. Soc., 75 (1953), 428–443 | DOI | MR | Zbl

[9] S. Armstrong, “Ricci-flat holonomy: a classification”, J. Geom. Phys., 57:6 (2007), 1457–1475 | DOI | MR | Zbl

[10] V. V. Astrakhantsev, “Holonomy groups of four-dimensional pseudo-Riemann spaces”, Math. Notes, 9:1 (1971), 33–37 | DOI | MR | Zbl

[11] A. Batrachenko, M. J. Duff, J. T. Liu, W. Y. Wen, “Generalized holonomy of M-theory vacua”, Nuclear Phys. B, 726:1-2 (2005), 275–293 | DOI | MR | Zbl

[12] H. Baum, “Conformal Killing spinors and the holonomy problem in Lorentzian geometry – a survey of new results”, Symmetries and overdetermined systems of partial differential equations, IMA Vol. Math. Appl., 144, Springer, New York, 2008, 251–264 | DOI | MR | Zbl

[13] H. Baum, “Holonomy groups of Lorentzian manifolds: a status report”, Global differential geometry, Springer Proc. Math., 17, Springer-Verlag, Berlin–Heidelberg, 2012, 163–200 | DOI | Zbl

[14] H. Baum, A. Juhl, Conformal differential geometry. $Q$-curvature and conformal holonomy, Oberwolfach Semin., 40, Birkhäuser Verlag, Basel, 2010, x+152 pp. | DOI | MR | Zbl

[15] H. Baum, I. Kath, “Parallel spinors and holonomy groups on pseudo-Riemannian spin manifolds”, Ann. Global Anal. Geom., 17:1 (1999), 1–17 | DOI | MR | Zbl

[16] H. Baum, K. Lärz, T. Leistner, “On the full holonomy group of special Lorentzian manifolds”, Math. Z., 277:3-4 (2014), 797–828 | DOI | MR | Zbl

[17] H. Baum, O. Müller, “Codazzi spinors and globally hyperbolic manifolds with special holonomy”, Math. Z., 258:1 (2008), 185–211 | DOI | MR | Zbl

[18] Ya. V. Bazaikin, “On the new examples of complete noncompact $\operatorname{Spin}(7)$-holonomy metrics”, Sib. Math. J., 48:1 (2007), 8–25 | DOI | MR | Zbl

[19] Ya. V. Bazaikin, “Globally hyperbolic Lorentzian spaces with special holonomy groups”, Sib. Math. J., 50:4 (2009), 567–579 | DOI | MR | Zbl

[20] Ya. V. Bazaikin, E. G. Malkovich, “$G_2$-holonomy metrics connected with a 3-Sasakian manifold”, Sib. Math. J., 49:1 (2008), 1–4 | DOI | MR | Zbl

[21] L. Bérard-Bergery, A. Ikemakhen, “On the holonomy of Lorentzian manifolds”, Differential geometry: geometry in mathematical physics and related topics (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., 54, Part 2, Amer. Math. Soc., Providence, RI, 1993, 27–40 | DOI | MR | Zbl

[22] L. Bérard-Bergery, A. Ikemakhen, “Sur l'holonomie des variétés pseudo-riemanniennes de signature $(n,n)$”, Bull. Soc. Math. France, 125:1 (1997), 93–114 | MR | Zbl

[23] M. Berger, “Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes”, Bull. Soc. Math. France, 83 (1955), 279–330 | MR | Zbl

[24] M. Berger, “Les espaces symétriques noncompacts”, Ann. Sci. École Norm. Sup. (3), 74 (1957), 85–177 | MR | Zbl

[25] A. L. Besse, Einstein manifolds, Ergeb. Math. Grenzgeb. (3), 10, Springer-Verlag, Berlin, 1987, xii+510 pp. | DOI | MR | MR | Zbl | Zbl

[26] N. I. Bezvitnaya, “Holonomy groups of pseudo-quaternionic-Kählerian manifolds of non-zero scalar curvature”, Ann. Global Anal. Geom., 39:1 (2011), 99–105 | DOI | MR | Zbl

[27] N. I. Bezvitnaya, “Holonomy algebras of pseudo-hyper-Kählerian manifolds of index 4”, Differential Geom. Appl., 31:2 (2013), 284–299 | DOI | MR | Zbl

[28] O. F. Blanco, M. Sánchez, J. M. M. Senovilla, “Complete classifcation of second-order symmetric spacetimes”, J. Phys. Conf. Ser., 229:1 (2010), 012021, 5 pp. | DOI

[29] O. F. Blanco, M. Sánchez, J. M. M. Senovilla, “Structure of second-order symmetric Lorentzian manifolds”, J. Eur. Math. Soc. (JEMS), 15:2 (2013), 595–634 | DOI | MR | Zbl

[30] A. Bolsinov, D. Tsonev, “On a new class class of holonomy groups in pseudo-Riemannian geometry”, J. Differential Geom., 97:3 (2014), 377–394 | MR | Zbl

[31] A. Borel, A. Lichnerowicz, “Groupes d'holonomie des variétés riemanniennes”, C. R. Acad. Sci. Paris, 234 (1952), 1835–1837 | MR | Zbl

[32] Ch. Boubel, “On the holonomy of Lorentzian metrics”, Ann. Fac. Sci. Toulouse Math. (6), 16:3 (2007), 427–475 | DOI | MR | Zbl

[33] Ch. Boubel, A. Zeghib, Dynamics of some Lie subgroups of $O(n,1)$, applications, Prépublication de l'ENS Lyon No 315, 2003 http://www.umpa.ens-lyon.fr/UMPA/Prepublications/

[34] C. P. Boyer, K. Galicki, “Sasakian geometry, holonomy, and supersymmetry”, Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc., Zürich, 2010, 39–83 | DOI | MR | Zbl

[35] J. Brannlund, A. Coley, S. Hervik, “Supersymmetry, holonomy and Kundt spacetimes”, Classical Quantum Gravity, 25:19 (2008), 195007, 10 pp. | DOI | MR | Zbl

[36] J. Brannlund, A. Coley, S. Hervik, “Holonomy, decomposability, and relativity”, Canad. J. Phys., 87:3 (2009), 241–243 | DOI

[37] M. Brozos-Vázquez, E. García-Río, P. Gilkey, S. Nikčević, R. Vázquez-Lorenzo, The geometry of Walker manifolds, Synth. Lect. Math. Stat., 5, Morgan Claypool Publishers, Williston, VT, 2009, xviii+159 pp. | MR | Zbl

[38] R. L. Bryant, “Metrics with exceptional holonomy”, Ann. of Math. (2), 126:3 (1987), 525–576 | DOI | MR | Zbl

[39] R. L. Bryant, “Classical, exceptional and exotic holonomies: a status report”, Actes de la table ronde de géométrie différentielle (Luminy, 1992), Sémin. Congr., 1, Soc. Math. France, Paris, 1996, 93–165 | MR | Zbl

[40] R. Bryant, “Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor”, Global analysis and harmonic analysis (Marseille–Luminy, 1999), Sémin. Congr., 4, Soc. Math. France, Paris, 2000, 53–94 | MR | Zbl

[41] M. Cahen, N. Wallach, “Lorentzian symmetric spaces”, Bull. Amer. Math. Soc., 76:3 (1970), 585–591 | DOI | MR | Zbl

[42] É. Cartan, “Les groupes réels simples finis et continus”, Ann. Sci. École Norm. Sup. (3), 31 (1914), 263–355 | MR | Zbl

[43] É. Cartan, “Sur une classe remarquable d'espaces de Riemann”, Bull. Soc. Math. France, 54 (1926), 214–264 | MR | Zbl

[44] É. Cartan, “Les groupes d'holonomie des espaces généralisés”, Acta. Math., 48:1-2 (1926), 1–42 | DOI | Zbl

[45] S. Cecotti, A geometric introduction to $F$-theory, Lectures Notes SISSA, 2010, 130 pp. http://people.sissa.it/~cecotti/FNOTES10.pdf

[46] A. Coley, A. Fuster, S. Hervik, “Supergravity solutions with constant scalar invariants”, Internat. J. Modern Phys. A, 24:6 (2009), 1119–1133 | DOI | MR | Zbl

[47] M. Cvetič, G. W. Gibbons, H. Lü, C. N. Pope, “New complete noncompact $\operatorname{Spin}(7)$ manifolds”, Nuclear Phys. B, 620:1-2 (2002), 29–54 | DOI | MR | Zbl

[48] G. de Rham, “Sur la réductibilité d'un espace de Riemann”, Comment. Math. Helv., 26 (1952), 328–344 | DOI | MR | Zbl

[49] A. Derdzinski, W. Roter, “The local structure of conformally symmetric manifolds”, Bull. Belg. Math. Soc. Simon Stevin, 16:1 (2009), 117–128 | MR | Zbl

[50] A. J. Di Scala, C. Olmos, “The geometry of homogeneous submanifolds of hyperbolic space”, Math. Z., 237:1 (2001), 199–209 | DOI | MR | Zbl

[51] D. V. Egorov, “QR-submanifolds and Riemannian metrics with holonomy $G_2$”, Math. Notes, 90:5 (2011), 763–766 | DOI | DOI | MR | Zbl

[52] J. M. Figueroa-O'Farrill, “Breaking the M-waves”, Classical Quantum Gravity, 17:15 (2000), 2925–2947 | DOI | MR | Zbl

[53] J. Figueroa-O'Farrill, P. Meessen, S. Philip, “Supersymmetry and homogeneity of M-theory backgrounds”, Classical Quantum Gravity, 22:1 (2005), 207–226 | DOI | MR | Zbl

[54] Th. Friedrich, “Zur Existenz paralleler Spinorfelder über Riemannschen Mannigfaltigkeiten”, Colloq. Math., 44:2 (1981), 277–290 | MR | Zbl

[55] A. S. Galaev, “The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds”, Differential Geom. Appl., 22:1 (2005), 1–18 | DOI | MR | Zbl

[56] A. S. Galaev, “Metrics that realize all Lorentzian holonomy algebras”, Int. J. Geom. Methods Mod. Phys., 3:5-6 (2006), 1025–1045 | DOI | MR | Zbl

[57] A. S. Galaev, “Isometry groups of Lobachevskian spaces, similarity transformation groups of Euclidean spaces and Lorentzian holonomy groups”, Rend. Circ. Mat. Palermo (2) Suppl., 2006, no. 79, 87–97 | MR | Zbl

[58] A. S. Galaev, Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2, PhD thesis, Humboldt Univ., Berlin, 2006, 135 pp. | Zbl

[59] A. S. Galaev, “One component of the curvature tensor of a Lorentzian manifold”, J. Geom. Phys., 60:6-8 (2010), 962–971 | DOI | MR | Zbl

[60] A. S. Galaev, “Holonomy of Einstein Lorentzian manifolds”, Classical Quantum Gravity, 27:7 (2010), 075008, 13 pp. | DOI | MR | Zbl

[61] A. S. Galaev, “On the Einstein equation on Lorentzian manifolds with parallel distributions of isotropic lines”, Fiziko-matematicheskie nauki, Uchen. zap. Kazan. gos. un-ta. Ser. Fiz.-matem. nauki, 153, no. 3, Izd-vo Kazan. un-ta, Kazan, 2011, 165–174

[62] A. S. Galaev, “Examples of Einstein spacetimes with recurrent null vector fields”, Classical Quantum Gravity, 28:17 (2011), 175022, 6 pp. | DOI | MR | Zbl

[63] A. S. Galaev, “Irreducible holonomy algebras of Riemannian supermanifolds”, Ann. Global Anal. Geom., 42:1 (2012), 1–27 | DOI | MR | Zbl

[64] A. Galaev, “Some applications of the Lorentzian holonomy algebras”, J. Geom. Symmetry Phys., 26 (2012), 13–31 | MR | Zbl

[65] A. S. Galaev, “Holonomy algebras of Einstein pseudo-Riemannian manifolds”, 2012, 16 pp., arXiv: 1206.6623

[66] A. S. Galaev, “Conformally flat Lorentzian manifolds with special holonomy groups”, Sb. Math., 204:9 (2013), 1264–1284 | DOI | DOI | MR | Zbl

[67] A. S. Galaev, “Pseudo-Riemannian manifolds with recurrent spinor fields”, Sib. Math. J., 54:4 (2013), 604–613 | DOI | MR | Zbl

[68] A. S. Galaev, “About the classification of the holonomy algebras of Lorentzian manifolds”, Sib. Math. J., 54:5 (2013), 798–804 | DOI | MR | Zbl

[69] A. S. Galaev, “Note on the holonomy groups of pseudo-Riemannian manifolds”, Math. Notes, 93:6 (2013), 810–815 | DOI | DOI | MR | Zbl

[70] A. S. Galaev, “On the de Rham–Wu decomposition for Riemannian and Lorentzian manifolds”, Classical Quantum Gravity, 31:13 (2014), 135007, 13 pp. | DOI | MR | Zbl

[71] A. S. Galaev, “How to find the holonomy algebra of a Lorentzian manifold”, Lett. Math. Phys., 105:2 (2015), 199–219 | DOI | MR | Zbl

[72] A. Galaev, T. Leistner, “Holonomy groups of Lorentzian manifolds: classification, examples, and applications”, Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, 2008, 53–96 | DOI | MR | Zbl

[73] A. S. Galaev, T. Leistner, “Recent developments in pseudo-Riemannian holonomy theory”, Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lect. Math. Theor. Phys., 16, Eur. Math. Soc., Zürich, 2010, 581–627 | DOI | MR | Zbl

[74] A. S. Galaev, T. Leistner, “On the local structure of Lorentzian Einstein manifolds with parallel distribution of null lines”, Classical Quantum Gravity, 27:22 (2010), 225003, 16 pp. | DOI | MR | Zbl

[75] R. Ghanam, G. Thompson, “Two special metrics with $R_{14}$-type holonomy”, Classical Quantum Gravity, 18:11 (2001), 2007–2014 | DOI | MR | Zbl

[76] G. W. Gibbons, C. N. Pope, “Time-dependent multi-centre solutions from new metrics with holonomy $\operatorname{Sim}(n-2)$”, Classical Quantum Gravity, 25:12 (2008), 125015, 21 pp. | DOI | MR | Zbl

[77] G. W. Gibbons, “Holonomy old and new”, Progr. Theoret. Phys. Suppl., 177 (2009), 33–41 | DOI | Zbl

[78] J. Grover, J. B. Gutowski, C. A. R. Herdeiro, P. Meessen, A. Palomo-Lozano, W. A. Sabra, “Gauduchon–Tod structures, Sim holonomy and De Sitter supergravity”, J. High Energy Phys., 7 (2009), 069, 20 pp. | DOI | MR

[79] S. S. Gubser, “Special holonomy in string theory and M-theory”, Strings, branes and extra dimensions, TASI 2001, World Sci. Publ., River Edge, NJ, 2004, 197–233 | DOI | MR | Zbl

[80] G. Hall, “Projective relatedness and conformal flatness”, Cent. Eur. J. Math., 10:5 (2012), 1763–1770 | DOI | MR | Zbl

[81] G. S. Hall, D. P. Lonie, “Holonomy groups and spacetimes”, Classical Quantum Gravity, 17:6 (2000), 1369–1382 | DOI | MR | Zbl

[82] S. Helgason, Differential geometry and symmetric spaces, Pure Appl. Math., XII, Academic Press, New York–London, 1962, xiv+486 pp. | MR | Zbl | Zbl

[83] N. Hitchin, “Harmonic spinors”, Adv. in Math., 14:1 (1974), 1–55 | DOI | MR | Zbl

[84] K. Honda, K. Tsukada, “Conformally flat semi-Riemannian manifolds with nilpotent Ricci operators and affine differential geometry”, Ann. Global Anal. Geom., 25:3 (2004), 253–275 | DOI | MR | Zbl

[85] A. Ikemakhen, “Sur l'holonomie des variétés pseudo-riemanniennes de signature $(2,2+n)$”, Publ. Mat., 43:1 (1999), 55–84 | DOI | MR | Zbl

[86] A. Ikemakhen, “Groupes d'holonomie et spineurs parallèles sur les variétés pseudo-riemanniennes complètement réductibles”, C. R. Math. Acad. Sci. Paris, 339:3 (2004), 203–208 | DOI | MR | Zbl

[87] T. Jacobson, J. D. Romano, “The spin holonomy group in general relativity”, Comm. Math. Phys., 155:2 (1993), 261–276 | DOI | MR | Zbl

[88] D. D. Joyce, Compact manifolds with special holonomy, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2000, xii+436 pp. | MR | Zbl

[89] D. Joyce, Riemannian holonomy groups and calibrated geometry, Oxf. Grad. Texts Math., 12, Oxford Univ. Press, Oxford, 2007, x+303 pp. | MR | Zbl

[90] V. R. Kaigorodov, “Structure of space-time curvature”, J. Soviet Math., 28:2 (1985), 256–273 | DOI | MR | Zbl

[91] R. P. Kerr, J. N. Goldberg, “Some applications of the infinitesimal-holonomy group to the Petrov classification of Einstein spaces”, J. Math. Phys., 2:3 (1961), 327–332 | DOI | MR | Zbl

[92] R. P. Kerr, J. N. Goldberg, “Einstein spaces with four-parameter holonomy groups”, J. Math. Phys., 2:3 (1961), 332–336 | DOI | MR | Zbl

[93] V. F. Kirichenko, “Conformally flat and locally conformal Kähler manifolds”, Math. Notes, 51:5 (1992), 462–468 | DOI | MR | Zbl

[94] S. Kobayashi, K. Nomizu, Foundations of differential geometry, v. I, II, Interscience Tracts in Pure and Applied Mathematics, Interscience Publishers John Wiley Sons, Inc., New York–London–Sydney, 1963, 1969, xi+329 pp., xv+470 pp. | MR | MR | MR | MR | Zbl | Zbl

[95] T. Krantz, “Kaluza–Klein-type metrics with special Lorentzian holonomy”, J. Geom. Phys., 60:1 (2010), 74–80 | DOI | MR | Zbl

[96] M. Kurita, “On the holonomy group of the conformally flat Riemannian manifold”, Nagoya Math. J., 9 (1955), 161–171 | MR | Zbl

[97] K. Lärz, Riemannian foliations and the topology of Lorentzian manifolds, 2010, 16 pp., arXiv: 1010.2194

[98] T. Leistner, “Lorentzian manifolds with special holonomy and parallel spinors”, Proceedings of the 21st winter school “Geometry and physics” (Srní, 2001), Rend. Circ. Mat. Palermo (2) Suppl., 2002, no. 69, 131–159 | MR | Zbl

[99] T. Leistner, Holonomy and parallel spinors in Lorentzian geometry, PhD thesis, Humboldt-Univ., Berlin, 2003, xii+178 pp. | Zbl

[100] T. Leistner, “On the classification of Lorentzian holonomy groups”, J. Differential Geom., 76:3 (2007), 423–484 | MR | Zbl

[101] T. Leistner, D. Schliebner, Completeness of compact Lorentzian manifolds with special holonomy, 2013, 30 pp., arXiv: 1306.0120

[102] J. Lewandowski, “Reduced holonomy group and Einstein equations with a cosmological constant”, Classical Quantum Gravity, 9:10 (1992), L147–L151 | DOI | MR | Zbl

[103] B. McInnes, “Methods of holonomy theory for Ricci-flat Riemannian manifolds”, J. Math. Phys., 32:4 (1991), 888–896 | DOI | MR | Zbl

[104] S. Merkulov, L. Schwachhöfer, “Classification of irreducible holonomies of torsion-free affine connections”, Ann. of Math. (2), 150:1 (1999), 77–149 | DOI | MR | Zbl

[105] O. Müller, M. Sánchez, “An invitation to Lorentzian geometry”, Jahresber. Dtsch. Math.-Ver., 115:3-4 (2014), 153–183 | DOI | MR | Zbl

[106] J. D. Norton, “Einstein, Nordström and the early demise of scalar, Lorentz-covariant theories of gravitation”, Arch. Hist. Exact Sci., 45:1 (1992), 17–94 | DOI | MR | Zbl

[107] C. Olmos, “A geometric proof of the Berger holonomy theorem”, Ann. of Math. (2), 161:1 (2005), 579–588 | DOI | MR | Zbl

[108] A. Z. Petrov, Einstein spaces, Pergamon Press, Oxford–Edinburgh–New York, 1969, xiii+411 pp. | MR | MR | Zbl | Zbl

[109] J. F. Schell, “Classification of four-dimensional Riemannian spaces”, J. Math. Phys., 2:2 (1961), 202–206 | DOI | MR | Zbl

[110] R. Schimming, “Riemannsche Räume mit ebenfrontiger und mit ebener Symmetrie”, Math. Nachr., 59:1-6 (1974), 129–162 | DOI | MR | Zbl

[111] L. J. Schwachhöfer, “Connections with irreducible holonomy representations”, Adv. Math., 160:1 (2001), 1–80 | DOI | MR | Zbl

[112] J. M. M. Senovilla, “Second-order symmetric Lorentzian manifolds. I. Characterization and general results”, Classical Quantum Gravity, 25:24 (2008), 245011, 25 pp. | DOI | MR | Zbl

[113] R. Hernández, K. Sfetsos, D. Zoakos, “Supersymmetry and Lorentzian holonomy in various dimensions”, J. High Energy Phys., 9 (2004), 010, 17 pp. | DOI | MR

[114] J. Simons, “On the transitivity of holonomy systems”, Ann. of Math. (2), 76:2 (1962), 213–234 | DOI | MR | Zbl

[115] V. V. Slavskii, “Conformally-flat metrics and pseudo-Euclidean geometry”, Sib. Math. J., 35:3 (1994), 605–613 | DOI | MR | Zbl

[116] R. S. Strichartz, “Linear algebra of curvature tensors and their covariant derivatives”, Canad. J. Math., 40:5 (1988), 1105–1143 | DOI | MR | Zbl

[117] S. Tanno, “Curvature tensors and covariant derivatives”, Ann. Mat. Pura Appl. (4), 96:1 (1973), 233–241 | DOI | MR | Zbl

[118] A. L. Onishchik, È. B. Vinberg, Lie groups and algebraic groups, Springer Ser. Soviet Math., Springer-Verlag, Berlin, 1990, xx+328 pp. | DOI | MR | MR | Zbl | Zbl

[119] V. P. Vizgin, Relyativistskaya teoriya tyagoteniya (istoki i formirovanie, 1900–1915), Nauka, M., 1981, 352 pp. | MR | Zbl

[120] A. G. Walker, “On parallel fields of partially null vector spaces”, Quart. J. Math., Oxford Ser., 20 (1949), 135–145 | DOI | MR | Zbl

[121] M. Y. Wang, “Parallel spinors and parallel forms”, Ann. Global Anal. Geom., 7:1 (1989), 59–68 | DOI | MR | Zbl

[122] H. Wu, “On the de Rham decomposition theorem”, Illinois J. Math., 8:2 (1964), 291–311 | MR | Zbl

[123] S.-T. Yau, “On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I”, Comm. Pure Appl. Math., 31:3 (1978), 339–411 | DOI | MR | Zbl