L-infinity Algebras From Multicontact Geometry

Abstract

I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on manifolds whose characteristic symmetries span a constant dimensional distribution. I call them pre-multicontact structures. Every distribution is almost everywhere, locally, a pre-multicontact structure. After showing that the standard symplectization of contact manifolds generalizes naturally to a (pre-)multisymplectization of (pre-)multicontact manifolds, I make use of results by C. Rogers and M. Zambon to associate a canonical $L_{\infty}$-algebra to any (pre-)multicontact structure. Such $L_{\infty}$-algebra is a multicontact version of the Jacobi bracket on a contact manifold. However, unlike the multisymplectic $L_\infty$-algebra of Rogers and Zambon, the multicontact $L_\infty$-algebra is always a homological resolution of a Lie algebra. Finally, I describe in local coordinates the $L_{\infty}$-algebra associated to the Cartan distribution on jet spaces.