no. 1
Une caractérisation des formes symplectiques
Annales de l'Institut Fourier, Tome 48 (1998) no. 1, pp. 265-280.

On montre qu’une 2-forme non nulle sur une variété M, telle que le pseudogroupe des difféomorphismes locaux la préservant soit transitif sur le fibré des directions tangentes, est symplectique si la dimension de M n’est pas 6. De plus, il y a un contre-exemple en dimension 6, dont on montre qu’il est essentiellement unique.

It is shown that a nonzero 2-form on a manifold M, such that the pseudogroup of local diffeomorphisms preserving it acts transitively on the bundle of tangent directions, is symplectic if dim M is not 6. Moreover, there is a counterexample in 6 dimensions, which is shown to be essentially unique.

@article{AIF_1998__48_1_265_0,
     author = {S\'evennec, Bruno},
     title = {Une caract\'erisation des formes symplectiques},
     journal = {Annales de l'Institut Fourier},
     pages = {265--280},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {1},
     year = {1998},
     doi = {10.5802/aif.1618},
     zbl = {0943.53047},
     mrnumber = {99b:53047},
     language = {fr},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1618/}
}
TY  - JOUR
AU  - Sévennec, Bruno
TI  - Une caractérisation des formes symplectiques
JO  - Annales de l'Institut Fourier
PY  - 1998
SP  - 265
EP  - 280
VL  - 48
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1618/
DO  - 10.5802/aif.1618
LA  - fr
ID  - AIF_1998__48_1_265_0
ER  - 
%0 Journal Article
%A Sévennec, Bruno
%T Une caractérisation des formes symplectiques
%J Annales de l'Institut Fourier
%D 1998
%P 265-280
%V 48
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1618/
%R 10.5802/aif.1618
%G fr
%F AIF_1998__48_1_265_0
Sévennec, Bruno. Une caractérisation des formes symplectiques. Annales de l'Institut Fourier, Tome 48 (1998) no. 1, pp. 265-280. doi : 10.5802/aif.1618. https://aif.centre-mersenne.org/articles/10.5802/aif.1618/

[Ar] V.I. Arnold, Méthodes mathématiques de la mécanique classique, Mir, 1976. | MR | Zbl

[Be] A. Besse, Einstein manifolds, Springer Verlag, 1987. | MR | Zbl

[Bo1] A. Borel, Some remarks about Lie groups transitive on spheres and tori, Bull. Amer. Math. Soc., 55 (1949), 580-587. | MR | Zbl

[Bo2] A. Borel, Le plan projectif des octaves et les sphères comme espaces homogènes, C. Rend. Acad. Sc., 230 (1950), 1378-1380. | MR | Zbl

[Br] R. Bryant, Submanifolds and special structures on the octonians, J. Differential Geometry, 17 (1982), 185-232. | MR | Zbl

[Ca] E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math. Soc., 87 (1958), 407-438. | MR | Zbl

[Ec1] B. Eckmann, Stetige Lösungen linearer Gleichungssysteme, Comment. Math. Helv., 15 (1943), 318-339. | MR | Zbl

[Ec2] B. Eckmann, Complex-analytic manifolds, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pp. 420-427, Amer. Math. Soc., Providence, R. I., 1952. | Zbl

[H] R. Hartshorne, Algebraic geometry, Springer Verlag, 1977. | MR | Zbl

[Ha] R. Harvey, Spinors and calibrations [ch. 6], Academic Press, 1990. | MR | Zbl

[He] S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, 1978. | MR | Zbl

[Hi] F. Hirzebruch, Topological methods in algebraic geometry [ch. 1, §§3,4], Springer Verlag, 1966. | MR | Zbl

[Ho] G. Hochschild, La structure des groupes de Lie, Dunod, 1968. | Zbl

[HoGS] H.H. Homer, W.D. Glover, R.E. Stong, Splitting the tangent bundle of projective space, Indiana Univ. Math. J., 31, No. 2 (1982), 161-166. | MR | Zbl

[Hu] D. Husemoller, Fiber bundles [ch. 17], Springer Verlag, 3ème éd., 1994.

[Ko] S. Kobayashi, Transformation groups in differential geometry, Springer Verlag, 1972. | MR | Zbl

[MiSt] J. Milnor, J. Stasheff, Characteristic classes, Princeton University Press, 1974. | MR | Zbl

[Mo] D. Montgomery, Simply connected homogeneous spaces, Proc. Amer. Math. Soc., 1 (1950), 467-469. | MR | Zbl

[MoSa] D. Montgomery, H. Samelson, Transformation groups of spheres, Ann. of Math., 44 (1943), 454-470. | MR | Zbl

[Mu] D. Mumford, Algebraic geometry I. Complex projective varieties, Springer Verlag, 1976. | Zbl

[On] A.L. Onischik, On Lie groups transitive on compact manifolds, I, II, III, Amer. Math. Soc. Translations, 73 (1968), 59-72; Mat. Sb., 116 (1967), 373-388; Mat. Sb., 117 (1968), 255-263. | Zbl

[On2] A.L. Onischik (Ed.), Lie groups and Lie algebras 1. Foundations of Lie theory, Lie transformations groups, Springer Verlag, 1993. | Zbl

[Sa] S. Salamon, Riemannian geometry and holonomy groups [ch. 10], Longman, 1989. | MR | Zbl

[Sz] Z.I. Szabo, A short topological proof for the symmetry of 2 point homogeneous spaces, Invent. Math., 106, No. 1 (1991), 61-64. | MR | Zbl

Cité par Sources :