symp-foliations

Here are my comments on leafwise symplectic foliations:

As is proved by David Martinez Torres, any closed manifold with codimension one 2-calibrated foliation has a closed transverse 3-dimensional submanifold which inherits a taut foliation with the “same” leaf space as the original one, i.e., which meets every leaf in a single connected symplectic submanifold transversely. From this point of view, you can prove many results in relation with those for taut foliations on 3-manifolds. Maybe you need specialists on taut foliation as well as those on holomorphic approximation.

Perhaps you know that Yoshihiko Mitsumatsu has found a leafwise symplectic structure of the Lawson foliation of the 5-sphere. Here the symplectic structure of the leaf is the restriction of a non-closed 2-form on the 5-sphere. Note that the leaf space of the Lawson foliation is homeomorphic to that of the Reeb foliation of the 3-sphere, which can be decomposed into two circles presenting the interior foliation of the Reeb components by removing a point presenting the closed leaf which is close to any other leaf. This contrasts to the fact that the leaf space of a taut foliation is a quotient of a single circle. I really want to prove that a Novikov type theorem holds for leafwise symplectic foliations. A possible formulation: Given a codimension one oriented foliation on a closed odd dimensional manifold, if the space of leafwise closed non-degenerate 2-forms is non-empty and contains no closed forms, there exists a closed leaf. (This holds in dimension 3.) Gael Meigniez could have some idea to prove it.

While my preprint (which includes a stupid but easily removable error in the statement of the key proposition) has not yet updated, I have proved that the standard contact structure on a 5-sphere can be deformed into the Lawson foliation with Mitsumatsu’s leafwise symplectic structure via a family of almost contact structures. This is also valid in a similar situation including S⁴×S¹. Anyway, my formulation is based on the study of twisted Jacobi structures, and this might be what I should I explain here. Twisted Jacobi structure is a singular foliation by twisted contact and twisted symplectic leaves. Twisted Poisson structure (or Poisson structure with back ground 3-form) is a twisted Jacobi structure without twisted contact leaves. A twisted contact structure is a pair of a 1-form α and a 2-form τ such that dα+τ is non-degenerate along \ker α. We call τ the twisting for non-degeneracy. Similarly, a closed 2-form becomes a twisted symplectic structure if a suitable twisting for non-degeneracy is added. Although I know that Eliashberg and Thurston discussed a possible generalization of their confoliation in higer dimensions in a different way, I would like to say that a 1-form α defines a confoliation if a linear family of twisting x τ for x∈(0,1] makes it twisted contact. Of course this implies that αΛ(dα)^n is non-negative. It is interesting that, in the case where α is already contact, and τ+dα is the boundary restriction of a symplectic form ω on a bordism of the contact manifold, the condition just says that the bordism is a weak symplectic filling. Anyway, the reason why I am really interested in the twisted setting is that a non-twisted Poisson structure (a singular foliation by simplectic leaves) is not a non-twisted Jacobi structure (a singular foliation by contact leaves). Thus we cannot consider our leafwise symplectic foliation and a (codimension zero) contact manifold simultaneously in the non-twisted setting. On the other hand, though the notion of twisted Jacobi structure includes that of twisted Poisson structure, it is highly general thing also known as Kirillov local Lie algebra. I imagine that this general thing as a metaphysical universe and the above family of twisting is a connection between classical and quantum physics which restrain us in a neighborhood of the real world. I also want to argue with the specialists on twisted Jacobi structure about the better definition of higher dimensional confoliations if the circumstance allows.