Let (M,g) be a riemannian manifold. I consider a flow over symplectic forms ω. I define a connection ∇ bounded with g and ω. The symmetric part of ∇ is defined by the fact that ∇ conserves g and the anti-symmetric part of ∇, by the fact that ω is conserved by ∇. The torsion of ∇ is T∇. Then I define a flow over ω by the equation:
∂ω∂t(X,Y)=ω(T∇(ei,X),T∇(ei,Y))
Have we solutions of the symplectic flow for short time?