Foliations with geometric structures

TL;DR

This thesis explores the classification and properties of foliations with specific geometric structures, such as conformal symplectic and contact structures, using advanced geometric and topological methods including the h-principle.

Contribution

It provides a classification of foliations with certain geometric structures on open manifolds and applies the h-principle to analyze foliations with contact leaves.

Findings

Classified foliations with locally conformal symplectic or contact leaves.

Applied h-principle to study foliations on contact manifolds.

Connected geometric structures to higher geometric frameworks like Poisson and Jacobi structures.

Abstract

A foliation on a manifold M can be informally thought of as a partition of M into injectively immersed submanifolds, called leaves. In this thesis we study foliations whose leaves carry some specific geometric structures. The thesis consists of two parts. In the first part we classify foliations on open manifolds whose leaves are either locally conformal symplectic or contact manifolds. These foliations can be described by some higher geometric structures - namely the Poisson and the Jacobi structures. In the second part of the thesis, we consider foliations on open contact manifolds whose leaves are contact submanifolds of the ambient space. Theory of h-principle plays the central role in deriving the main results of the thesis. It is a theory rich in topological techniques to solve partial differential relations which arise in connection with topology and geometry. All the geometric structures mentioned above satisfy some differential conditions and that brings us into the realm of the h-principle theory.