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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. ==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 co-ordinates. That is, : In the chart , the stripes 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 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 as , where , is homeomorphic to the plaques, and the points of parametrize the plaques in . If we pick in , then 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 global transversal sections of the foliation might not exist. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Foliation」の詳細全文を読む スポンサード リンク
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