Une caractérisation des formes symplectiques. (A characterization of symplectic forms.) (Q1265662)

For a given symplectic manifold \((M,\omega)\), the pseudogroup \(Ps(\omega)\), the set of local diffeomorphims of \(M\) preserving the form \(\omega\), acts transitively on \(TM \backslash 0\) and a fortiori on its projectivization \(P(TM)\). The main theorem of this paper is the converse of this statement. In other words, the present paper proves that, for a given manifold \((M,\omega)\) with \(\omega\) a non-degenerate two-form, if the pseudogroup \(Ps(\omega)\) acts transitively, then \(\omega\) must be closed, e.g., symplectic. The author first proves that if \(Ps(\omega)\) acts transitively on \(P(TM)\), it must either be symplectic or it must be contained in the pseudogroup of isometries of the sphere \(S^6\) with the unique nondegenerate two-form \(\omega_{S^6}\) under the compact exceptional Lie group \(G_2 \subset SO(7)\). The main theorem follows from this because it is obvious that the isometry group of a Riemannian metric cannot act transitively on \(TM\backslash 0\).