Abstract:

Contact structures are ubiquitous structures in differential geometry, topology, differential equations, and physics. Such structures have been well-studied since the late 1800s in some form or another. A classical result of Pfafff and Darboux demonstrates that, essentially, all contact structures are locally indistinguishable. That is, there are no local invariants under local diffeomorphisms that preserve a given contact structure.

For pairs of contact structures (bi-contact structures) such that each contact structure is simultaneously preserved by a diffeomorphism  (bicontactomorphisms), we uncover local differential invariants. These invariants arise from Cartan's method of equivalence and we discover families of local normal forms in special cases. This approach leads to natural generalizations of the contact circles introduced by H. Geiges and J. Gonzalo in the 1990s and more recently that of contact hyperbolas of D. Perrone. Time permitting, I will cover applications to some submanifold structures and characterizations of Beltrami fields.