Classification of $k$-forms on $\mathbb{R}^n$ and the existence of associated geometry on manifolds

Hông Vân Lê ; J. Vanžura

Čebyševskij sbornik, Tome 21 (2020) no. 2, pp. 362-382.

Résumé

In this paper we survey methods and results of classification of $k$-forms (resp. $k$-vectors on $\mathbb{R}^n$), understood as description of the orbit space of the standard $\mathrm{GL}(n, \mathbb{R})$-action on $\Lambda^k \mathbb{R}^{n*}$ (resp. on $\Lambda ^k \mathbb{R}^n$). We discuss the existence of related geometry defined by differential forms on smooth manifolds. This paper also contains an Appendix by Mikhail Borovoi on Galois cohomology methods for finding real forms of complex orbits.

Détail

Keywords: $ \mathrm {GL} (n, {\mathbb R})$-orbits in $\Lambda^k\mathbb{R}^{n*}$, $\theta$-group, geometry defined by differential forms, Galois cohomology.

Bibliographie

[1] Antonyan L. V., “Classification of 4-vectors on eight-dimensional space”, Proc. Seminar Vekt. and tensor. Analysis, 20, 1981, 144–161

[2] Antonyan L. V., Elashvili A. G., “Classification of spinors in dimension 16”, Proceedings of the Tbilisi institute of mathematics, LXX, 1982, 5–23

[3] A. Borel, Harish-Chandra, “Arithmetic subgroups of algebraic groups”, Annals of Math., 75 (1962), 485–535 | DOI | MR | Zbl

[4] M. Borovoi, W.A. de Graaf, H. V. Lê, Classification of 3-vectors on $\mathbb{R}^9$, in preparation

[5] M. Borovoi, “Abelianization of the second nonabelian Galois cohomology”, Duke Math. Journal, 72 (1993), 217–239 | DOI | MR | Zbl

[6] J. Bureš, J. Vanžura, “Multisymplectic forms of degree three in dimension seven”, Proceedings of the 22nd Winter School “Geometry and Physics” (Circolo Matematico di Palermo, Palermo), Rendiconti del Circolo Matematico di Palermo, Serie II, Supplemento, no. 71, 2003, 73–91 | MR | Zbl

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

[8] A. Cohen, A. Helminck, “Trilinear alternating forms on a vector space of dimension 7”, Commun. Algebra, 16 (1988), 1–25 | DOI | MR | Zbl

[9] F. Connolly, H. V. Lê and K. Ono, “Almost complex structures which are compatible with Kähler or symplectic structures”, Annals of Global Analysis and Geometry, 15 (1997), 325–334 | DOI | MR | Zbl

[10] J. Dieudonné, La géométrie des grouppes classiques, Springer, 1955 | MR

[11] D. J. Djokovic, “Classification of Trivectors of an Eight-dimensional Real Vector Space”, Linear and Multilinear Algebra, 13 (1983), 3–39 | DOI | MR | Zbl

[12] H. Dietrich, P. Faccin, W. A. de Graaf, “Regular subalgebras and nilpotent orbits of real graded Lie algebras”, Journal of algebra, 423 (2015), 1044–1079 | DOI | MR | Zbl

[13] Y. Eliashberg, N. Mishachev, Introduction to the h-Principle, AMS, 2002 | MR | Zbl

[14] J. J. Figueroa-O'Farrill, A. Santi, “Spencer Cohomology and 11-Dimensional Supergravity”, Commun. Math. Phys., 349 (2017), 627–660 | DOI | MR | Zbl

[15] J. J. Figueroa-O'Farrill, A. Santi, On the algebraic structure of Killing suprealgebras, arXiv: 1608.05915 | MR

[16] Th. Friedrich, I. Kath, A. Moroianu, U. Semmelmann, “On nearly parallel $G_2$-manifolds”, Journal Geom. Phys., 23 (1997), 259–286 | DOI | MR | Zbl

[17] D. Fiorenza, H. V. Lê, L. Schwachhöfer, L. Vitagliano, Strongly homotopy Lie algebras and deformation of claibrated submanifolds, arXiv: 1804.05732

[18] M. Gromov, “Pseudo holomorphic curves in symplectic manifolds”, Invent. Math., 82 (1985), 307–347 | DOI | MR | Zbl

[19] M. Gromov, Partial differential relations, Springer, Berlin, 1986 | MR | Zbl

[20] A. Gray, “Vector cross products on manifolds”, TAMS, 141 (1969), 465–504 ; Errata in: TAMS, 148 (1970), 625 | DOI | MR | Zbl | Zbl

[21] Gurevich G. B., “Classification des trivecteurs ayant le rang huit”, Dokl. Akad. Nauk SSSR, II:5–6 (1935), 51–113 | Zbl

[22] Gurevich G. B., “L'algebre du trivecteur”, Trudy Sem. Vekt. Tenz. Analizu, II–III, 1935, 355–356 | Zbl

[23] Gurevich G. B., Foundations of the theory of algebraic invariants, Noordhoff Ltd., Groningen, The Netherlands, 1964 | MR | MR | Zbl

[24] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978 | MR | Zbl

[25] N. Hitchin, “The geometry of three-forms in 6 and 7 dimensions”, J.D.G, 55 (2000), 547–576 | MR | Zbl

[26] N. Hitchin, “Stable forms and special metrics”, Contemporary math., 288, 2001, 70–89 | DOI | MR | Zbl

[27] R. Harvey, H. B. Lawson, “Calibrated geometries”, Acta Math., 182, 47–157 | MR

[28] N. Jacobson, Lie algebras, Dover, New York, 1979 | MR

[29] D. D. Joyce, Compact manifolds with Special Holonomy, Oxford University Press, 2000 | MR | Zbl

[30] D. Joyce, Riemannian Holonomy Groups and Calibrated Geometry, Oxford, 2007 | MR | Zbl

[31] Kac V.G., “Automorphisms of finite order of semisimple Lie algebras”, Funct. Anal. Appl., 3 (1969), 252–254 | DOI | MR | Zbl

[32] K. Kawai, H. V. Lê, L. Schwachhöfer, “The Frölicher-Nijenhuis bracket and the geometry of $G_2$- and Spin(7)-manifolds”, Ann. Math. Pur. Appl., 197 (2018), 411–432 | DOI | MR | Zbl

[33] H. V. Lê, Manifolds admitting a $\tilde G_2$-structure, arXiv: 0704.0503

[34] H. V. Lê, “Orbits in real $\mathbb{Z}_m$-graded semisimple Lie algebras”, J. Lie Theory, 21:2 (2011), 285–305 | MR | Zbl

[35] H.V. Lê, M. Munir, “Classification of compact homogeneous spaces with invariant $G_2$-structures”, Advances in Geometry, 12 (2012), 303–328 | MR | Zbl

[36] H. V. Lê, “Geometric structures associated with a simple Cartan 3-form”, Journal of Geometry and Physics, 70 (2013), 205–223 | DOI | MR | Zbl

[37] Lê H.V., Fomenko A.T., “A criterion for the minimality of Lagrangian submanifolds in Kählerian manifolds”, Math. Notes, 42 (1987), 810–816 | DOI | MR | Zbl

[38] H. V. Lê, M. Panak, J. Vanžura, “Manifolds admitting stable forms”, Comm. Math. Carolinae, 49:1 (2008), 101–117 | MR | Zbl

[39] H. V. Lê, J. Vanžura, “Cohomology theories on locally conformal symplectic manifolds”, Asian J. of Math., 19 (2015), 045–082 | DOI | MR

[40] H. V. Lê, J. Vanžura, “McLean's second variation formula revisited”, Journal of Geometry and Physics, 113 (2017), 188–196 | DOI | MR | Zbl

[41] J. Matinet, “Sur les singularités des formes différentielles”, Annales de l'institut Fourier, 20 (1970), 95–178 | DOI | MR

[42] N. Midoune, Classification des formes trilineaires alternees de rang 8 sur les corps finis, PhD Thesis, Université de Batna, 2009

[43] N. Midoune, L. Noui, “Trilinear alternating forms on a vector space of dimension 8 over a finite field”, Linear and Multilinear Algebra, 61 (2013), 15–21 | DOI | MR | Zbl

[44] L. Noui, “Transvecteurs de rang 8 sur un corps algebriquement clos”, C. R. Acad. Sci. Paris, Serie I: Algebre, 324 (1997), 611–614 | DOI | MR | Zbl

[45] L. Noui, Ph. Revoy, “Formes multilinaires alternes”, Ann. Math. Plaise Pascal, 1:2 (1994), 43–69 | DOI | MR | Zbl

[46] L. Noui, Ph. Revoy, “Algebres de lie orthogonales et formes trilineaires alternees”, Communications in algebra, 25 (1997), 617–622 | DOI | MR | Zbl

[47] T. Oshima, T. Matsuki, “Orbits on affine symmetric spaces under the action of the isotropy subgroups”, J. Math. Soc. Japan, 32:2 (1980), 399–414 | DOI | MR | Zbl

[48] Popov V. L., Vinberg E. B., “Invariant theory”, Algebraic geometry. IV, translated from 1989 Russian edition, Encyclopaedia of Mathematical Sciences, 55, Springer-Verlag, Berlin, 1994 | MR

[49] E.M. Rains, J. A. Sloane, “Self-dual codes”, Handbook of Coding Theory, eds. V.S. Pless, W. C. Huffman, Elsevier, Amsterdam, 1998, 177–294 | MR | Zbl

[50] W. Reichel, Über die trilinearen alternierenden Formen in 6 und 7 Veränderlichen, Dissertation, Greiswald, 1907 | Zbl

[51] Ph. Revoy, “Trivecteurs de rang 6”, Bull. Soc. Math. France Memoire, 59 (1979), 141–155 | DOI | MR | Zbl

[52] L. P. Rothschild, “Orbits in a real Reductive Lie algebra”, Trans. AMS, 168 (1972), 403–421 | DOI | MR | Zbl

[53] L. Ryvkin, Linear orbits of alternating forms on real vector spaces, arXiv: 1609.02184

[54] T. Salac, “Multisyplectic forms on 7-dimensional manifolds”, Differential Geometry and its Applications, 58 (2018), 120–140 | DOI | MR | Zbl

[55] S.M. Salamon, Riemannian geometry and holonomy groups, Pitman Res. Notes in Math., 201, Longman, Harlow, 1989 | MR | Zbl

[56] N. E. Steenrod, The topology of fibre bundles, Princeton University Press, 1951 | MR | Zbl

[57] J. A. Schouten, “Klassifizierung der alternierdenen Grössen dritten Grades in 7 Dimensionen”, Rend. Circ. mat. Palermo, 55 (1931), 137–156 | DOI

[58] J. P. Serre, Galois cohomology, corrected 2nd printing, Springer, 1997 | MR | Zbl

[59] C.H. Taubes, “More constraints on symplectic forms from Seiberg-Witten invariants”, Math. Res. Lett., 2 (1995), 9–14 | DOI | MR

[60] I. Vaisman, “Geometric quantization on presymplectic manifolds”, Monatshefte für Mathematik, 96 (1983), 293–310 | DOI | MR | Zbl

[61] Vinberg E. B., “On linear groups connected with periodic automorphisms of semisimple algebraic groups”, Sov. Math., Dokl., 16 (1976), 406–409 | MR | Zbl

[62] Vinberg E. B., “On the classification of nilpotent elements of a semisimple graded Lie algebras”, Sov. Math., Dokl., 16 (1976), 1517–1520 | MR | Zbl

[63] Vinberg E. B., “The Weyl group of a graded Lie algebra”, Math. USSR-Izv., 10 (1977), 463–495 (Russian) | DOI | MR | Zbl

[64] Vinberg E. B., “A classification of homogeneous elements of a semisimple graded Lie algebra”, Selecta Math. Sov., 6 (1987), 15–35 | Zbl | Zbl

[65] Vinberg E. B., Elashvili A. G., “A classification of the trivectors of nine-dimensional space”, Selecta Math. Sov., 7 (1988), 63–98 | MR | Zbl

[66] J. A. Wolf, A. Grey, “Homogeneous spaces defined by Lie group automorphisms. I; II”, JDG, 2(168), 77–114 | MR | Zbl

[67] R. Westwick, “Real trivectors of rank seven”, Lin. and multilin. Algebra, 10:3 (1981), 183–204 | DOI | MR | Zbl

[68] F. Witt, Special metric structures and closed forms, Ph.D. Thesis, University Oxford, 2004, arXiv: math.DG/0502443 | Zbl