| Title:
|
Manifolds admitting stable forms (English) |
| Author:
|
Lê, Hông-Van |
| Author:
|
Panák, Martin |
| Author:
|
Vanžura, Jiří |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
49 |
| Issue:
|
1 |
| Year:
|
2008 |
| Pages:
|
101-117 |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this note we give a direct method to classify all stable forms on $\Bbb R^n$ as well as to determine their automorphism groups. We show that in dimensions 6, 7, 8 stable forms coincide with non-degenerate forms. We present necessary conditions and sufficient conditions for a manifold to admit a stable form. We also discuss rich properties of the geometry of such manifolds. (English) |
| Keyword:
|
stable forms |
| Keyword:
|
automorphism groups |
| MSC:
|
53C10 |
| MSC:
|
53C15 |
| MSC:
|
53C25 |
| idZBL:
|
Zbl 1212.53051 |
| idMR:
|
MR2433628 |
| . |
| Date available:
|
2009-05-05T17:06:48Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119705 |
| . |
| Reference:
|
[1] Borel A., Chandra H.: Arithmetic subgroups of algebraic groups.Ann. of Math. 75 (1962), 485-535. Zbl 0107.14804, MR 0147566, 10.2307/1970210 |
| Reference:
|
[2] Bureš J., Vanžura J.: Multisymplectic forms of degree three in dimension seven.Rend. Circ. Mat. Palermo (2) Suppl. no. 71 (2003), 73-91. Zbl 1045.53017, MR 1982435 |
| Reference:
|
[3] Bryant R.: Metrics with exceptional holonomy.Ann. of Math. (2) 126 (1987), 525-576. Zbl 0637.53042, MR 0916718 |
| Reference:
|
[4] Bryant R.: Conformal geometry and $3$-plane fields on $6$-manifolds.arXiv:math.DG/0511110. MR 0974338 |
| Reference:
|
[5] Čadek M., Crabb M., Vanžura J.: Obstruction theory on $8$-manifolds.preprint 2007. |
| Reference:
|
[6] Djokovic D.Z.: Classification of trivectors of an eight-dimensional real vector space.Linear and Multilinear Algebra 13 (1983), 3-39. Zbl 0515.15011, MR 0691457, 10.1080/03081088308817501 |
| Reference:
|
[7] Dupont J.: $K$-theory obstructions to the existence of vector fields.Acta Math. 133 (1974), 67-80. Zbl 0313.57012, MR 0425980, 10.1007/BF02392142 |
| Reference:
|
[8] Gauntlett J.P., Martelli D., Pakis S., Waldram D.: $G$-structures and wrapped NS5-branes.Comm. Math. Phys. 247 (2004), 421-445, hep-th/0205050. MR 2063267, 10.1007/s00220-004-1066-y |
| Reference:
|
[9] Gray A.: Vector cross products on manifolds.Trans. Amer. Math. Soc. 141 (1969), 465-504; (Errata in Trans. Amer. Math. Soc. 148 (1970), 625). Zbl 0182.24603, MR 0243469, 10.1090/S0002-9947-1969-0243469-5 |
| Reference:
|
[10] Hitchin N.: The geometry of three-forms in six dimensions.J. Differential Geom. 55 (2000), 547-576. Zbl 1036.53042, MR 1863733 |
| Reference:
|
[11] Hitchin N.: Stable forms and special metrics.Contemp. Math. 288 (2001), 70-89. Zbl 1004.53034, MR 1871001, 10.1090/conm/288/04818 |
| Reference:
|
[12] Joyce D.: Compact Manifolds with Special Holonomy.Oxford University Press, Oxford, 2000. Zbl 1027.53052, MR 1787733 |
| Reference:
|
[13] Sato M., Kimura T.: A classification of irreducible prehomogeneous vector spaces and their relative invariants.Nagoya Math. J. 65 (1977), 1-155. MR 0430336 |
| Reference:
|
[14] Le H.V.: The existence of symplectic $3$-forms on $7$-manifolds.arXiv:math.DG/0603182. |
| Reference:
|
[15] Le H.V.: Manifolds admitting a $\tilde G_2$-structure.arXiv:07040503. |
| Reference:
|
[16] Murakami S.: On the automorphism of a real semi-simple Lie algebra.J. Math. Soc. Japan 4 (1952), 103-133. MR 0051829, 10.2969/jmsj/00420103 |
| Reference:
|
[17] Sagle A.: Malcev algebras.Trans. Amer. Math. Soc. 101 (1961), 426-458. Zbl 0101.02302, MR 0143791, 10.1090/S0002-9947-1961-0143791-X |
| Reference:
|
[18] Thomas E.: Fields of tangent $k$-planes on manifolds.Invent. Math. 3 (1967), 334-347. Zbl 0162.55402, MR 0217814, 10.1007/BF01402957 |
| Reference:
|
[19] Thomas E.: Vector fields on low dimensional manifolds.Math. Z. 103 (1967), 85-93. MR 0224109, 10.1007/BF01110620 |
| Reference:
|
[20] Tsimpis D.: M-theory on eight-manifolds revisited: $N=1$ supersymmetry and generalized Spin $(7)$-structures.preprint MPP-2005-129. MR 2219075 |
| Reference:
|
[21] Witt F.: Special metric structures and closed forms.Oxford Ph.D. Thesis, arXiv:math.DG/0502443. |
| Reference:
|
[22] Yamaguchi K.: Differential systems associated with simple graded Lie algebras.Adv. Stud. Pure Math. 22 (1993), 413-494. Zbl 0812.17018, MR 1274961 |
| . |
