Legendrians and coLegendrians#
Legendrians and coLegendrians are probably two of the most important classes of submanifolds in contact topology, and might also play a key role in the eventual understanding of contact structures in general.
On the one hand, Legendrian submanifolds are extensively studied both in dimension \(3\) and higher, mostly through the technique of holomorphic curves. As explained by M. Gromov in [Gro85], such technique distinguishes contact (and symplectic) topology from usual smooth topology, and therefore establishes them as independent subjects. However, it most likely won’t lead you to any actual understanding of the subject due to the abstract nature of the argument.
CoLegendrians, on the other hand, are studied by nobody but myself. They are rather special submanifolds which are \(1\) dimension higher than the Legendrians and are foliated by Legendrians in a generally singular manner. In a sense, it’s a natural generalization of the idea of studying Legendrian knots via bounding surfaces in dimension \(3\), which were extensively explored by pioneers of contact topology. But, as we’ll see, coLegendrians in higher dimensions behave drastically different from their \(2\)-dimensional siblings.
The hope is to understand Legendrians via the interplay with coLegendrians, but of course, the chance of success is slim but better than nothing. We’ll focus on contact manifolds of dimensions at least \(5\) but sometimes we’ll also touch upon contact \(3\)-manifolds because the number is, after all, close to \(5\).