Foliation - Definition

Definition

More formally, a dimension foliation of an -dimensional manifold is a covering by charts together with maps

such that for overlapping pairs the transition functions defined by

take the form

where denotes the first coordinates, and denotes the last p co-ordinates. That is,

and

In the chart, the stripes constant match up with the stripes on other charts . Technically, these stripes are called plaques of the foliation. In each chart, the plaques are dimensional submanifolds. These submanifolds piece together from chart to chart to form maximal connected injectively immersed submanifolds called the leaves of the foliation.

The notion of leaves allows for a more intuitive way of thinking about a foliation. A -dimensional foliation of an -manifold may be thought of as simply a collection of pairwise-disjoint, connected, immersed -dimensional submanifolds (the leaves of the foliation) of, such that for every point in, there is a chart with homeomorphic to containing such that for every leaf, meets in either the empty set or a countable collection of subspaces whose images under in are -dimensional affine subspaces whose first coordinates are constant.

If we shrink the chart it can be written in the form, where

and ,

and is homeomorphic to the plaques and the points of parametrize the plaques in . If we pick a

,

is a submanifold of that intersects every plaque exactly once. This is called a local transversal section of the foliation. Note that due to monodromy there might not exist global transversal sections of the foliation.