Integrability of geometric structures with respect to a Lie algebra

Compared to many other geometric structures, contact structures exhibit a peculiar behaviour: on the one hand, they are, intuitively, "flat", due to the existence of Darboux charts; on the other hand, they have an underlying $G$-structure which is not flat (its integrable version would be a symplectic foliation). To make this observation rigorous, Albert and Molino proposed in 1984 a different definition of integrability, "twisted" by the Heisenberg Lie algebra. Their point of view seems quite ad hoc and far from being satisfactory; in this talk we will understand that such phenomenon has roots in the world of transitive pseudogroups.

Indeed, the pseudogroup of local symmetries of the trivial $G$-structure on $\mathbb{R}^n$ is transitive because it contains translations (i.e. transformations by the action of the abelian group $(\mathbb{R}^n,+)$ ), while the pseudogroup of local symmetries of the canonical contact structure on $\mathbb{R}^n$ is transitive for a different reason: it contains transformations by the action of the Heisenberg group.

In order to clarify the role played by these two Lie groups (the abelian and the Heisenberg one), we have to zoom out and adopt the framework of Cartan bundles (principal bundles with special vector-valued 1-forms), which in turn arises from a more general framework of geometric structures described by any (possibly non transitive) pseudogroup. Then we introduce the new notions of Cartan type-extension (a Lie algebra extending in a suitable way the coefficients of the 1-form of a Cartan bundle) and of integrability with respect to them. The integrability with respect to, respectively, the abelian Lie algebra or the Heisenberg Lie algebra will yield precisely the integrability conditions for symplectic foliations and for contact structures. This approach provides also more insights on the integrability of Cartan geometries and of higher order G-structures.

This is based on joint works with Luca Accornero, Marius Crainic and Maria Amelia Salazar.