In the previous episode, we asked: if you have a family of foliations on a surface, do they arise as the movie of characteristic foliations of a contact structure? In this episode, we ask how unique these contact structures are.
We’ve seen that convex surfaces have wonderful foliations. We’re now going to consider the relationship between these foliations on surfaces, and contact structures
Starting from a Liouville 1-form on a surface, we have been led to 3-dimensional contact geometry, and convex surfaces. We now go in the other direction.
(Technical) We’re going to take Liouville structures and move them into 3 dimensions, to obtain contact structures.
(Technical) I’d like to show you some very nice geometry, involving some vector fields and differential forms.
Fun with a number, or two, or four. An interesting complex complexity.
Basic results and the power of convex surfaces.
Details on the significance of the paper, overtwisted discs, characteristic foliations, and contact structures.
My attempt to flesh out a few of the details.
Based on Casson and Bleiler’s book.