Une caractérisation des formes symplectiques

Sévennec, Bruno

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.

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