Foliation
Introduction
In mathematics, particularly in differential geometry and topology, a **foliation** is a geometric structure on a manifold that generalizes the notion of a fibration. Foliations are used to study the global structure of manifolds by decomposing them into simpler, well-understood pieces called leaves. Each leaf is a submanifold, and the collection of leaves forms a partition of the manifold.
Definition and Basic Properties
A foliation of a manifold \( M \) is a decomposition of \( M \) into a union of disjoint connected submanifolds, called leaves, which locally resemble the parallel planes in \(\mathbb{R}^n\). Formally, a \( p \)-dimensional foliation \( \mathcal{F} \) on an \( n \)-dimensional manifold \( M \) is given by an atlas of charts \(\{(U_i, \phi_i)\}\) such that:
1. Each \( U_i \) is an open subset of \( M \). 2. Each \( \phi_i: U_i \rightarrow \mathbb{R}^n \) is a homeomorphism. 3. The transition maps \( \phi_j \circ \phi_i^{-1} \) preserve the foliation structure, i.e., they map plaques (local pieces of leaves) to plaques.
The leaves of the foliation are the maximal connected submanifolds of \( M \) that are locally given by the level sets of the first \( p \) coordinates in the charts.
Examples of Foliations
Linear Foliations
A simple example of a foliation is given by the level sets of a smooth function \( f: M \rightarrow \mathbb{R} \). If \( f \) is a submersion, then the level sets \( f^{-1}(c) \) for \( c \in \mathbb{R} \) form a foliation of \( M \).
Foliations by Parallel Planes
Consider \( \mathbb{R}^3 \) with the standard coordinates \( (x, y, z) \). The planes \( z = c \) for \( c \in \mathbb{R} \) form a foliation of \( \mathbb{R}^3 \). Each leaf is a plane parallel to the \( xy \)-plane.
Holonomy and Transverse Structure
The concept of holonomy is crucial in the study of foliations. Holonomy describes how a leaf can twist and turn as one moves along it. Formally, the holonomy group of a leaf is the group of diffeomorphisms of a transverse section induced by loops in the leaf.
A transverse structure to a foliation is a geometric structure on the space of leaves. For example, a transverse measure assigns a measure to each transverse section in a way that is invariant under holonomy.
Foliations and Lie Groupoids
Foliations can be described using the language of Lie groupoids and Lie algebroids. A Lie groupoid is a category where every morphism is invertible, and both the objects and morphisms form smooth manifolds. The space of leaves of a foliation can be seen as the quotient of the manifold by the equivalence relation defined by the leaves, and this quotient can be described as a Lie groupoid.
Reeb Foliations
A Reeb foliation is a specific type of foliation on the 3-sphere \( S^3 \). It is constructed by taking the product of a circle with a Reeb component, which is a solid torus foliated in a specific way. Reeb foliations are important in the study of 3-manifolds and contact geometry.
Foliations and Dynamical Systems
Foliations are closely related to the study of dynamical systems. The leaves of a foliation can be seen as the trajectories of a dynamical system. In particular, the stable and unstable manifolds of a hyperbolic fixed point form a foliation of the phase space.
Applications of Foliations
Foliations have applications in various areas of mathematics and theoretical physics. In geometry, they are used to study the global structure of manifolds. In topology, foliations provide tools for understanding the classification of manifolds. In physics, foliations are used in the study of spacetime in general relativity and in the formulation of gauge theories.