Foliation

In mathematics, a foliation is a geometric tool for understanding manifolds. The leaves of a foliation consist of integrable subbundles of the tangent bundle. Foliating a manifold may split it up into pieces that interact simply. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension.

More formally, a dimension $p$ foliation $F$ of an $n$ -dimensional manifold $M$ is a covering by charts $U i$ together with maps

such that for overlapping pairs $U i, U j$ the transition functions $φ ij : R n \to R n$ defined by

take the form

where $x$ denotes the first $n - p$ coordinates, and $y$ denotes the last $p$ co-ordinates. That is,

In the chart $U i$ , the stripes $x = constant$ match up with the stripes on other charts $U j$ . Technically, these stripes are called plaques of the foliation. In each chart, the plaques are $p$ 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 $p$ -dimensional foliation of an $n$ -manifold $M$ may be thought of as simply a collection ${M a}$ of pairwise-disjoint, connected, immersed $p$ -dimensional submanifolds (the leaves of the foliation) of $M$ , such that for every point $x$ in $M$ , there is a chart $(U,\phi )$ $(U,\phi)$ with $U$ homeomorphic to $R n$ containing $x$ such that every leaf, $M a$ , meets $U$ in either the empty set or a countable collection of subspaces whose images under $\phi$ $\phi$ in $\phi (M_{a}\cap U)$ $\phi (M_a \cap U)$ are $p$ -dimensional affine subspaces whose first $n - p$ coordinates are constant.

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