Georgia Institute of Technology College of Sciences

It is a classical problem to study whether the h-principle holds for certain classes of maximally non-integrable distributions. The most studied case is that of contact structures, where there is a rich interplay between flexibility and rigidity, exemplified by the overtwisted vs tight dichotomy. For other types of maximally non-integrable distributions, no examples of rigidity are currently known.

In this talk I will discuss rigidity phenomena for fat distributions, which can be viewed as higher corank generalizations of contact structures. These admit natural symplectizations and contactizations. I will introduce a natural class of submanifolds in fat manifolds, called prelegendrians, which admit canonical Legendrian lifts to the contactization. The main result of the talk is that these submanifolds exhibit rigidity: in the “standard corank-2 fat manifold” there exists an infinite family of prelegendrian tori, all of them formally equivalent but pairwise not prelegendrian isotopic. In other words, the h-principle fails for prelegendrians. The talk is based on joint work with Álvaro del Pino and Wei Zhou.