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distribution
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(Definition)
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In the following we will mean $C^\infty$ when we say smooth.
Definition 1 Let $M$ be a smooth manifold of dimension $m$ Let $n \leq m$ and for each $x \in M$ we assign an $n$ dimensional subspace $\Delta_x \subset T_x(M)$ of the tangent space in such a way that for a neighbourhood $N_x \subset M$ of $x$ there exist $n$ linearly independent smooth vector fields $X_1,\ldots,X_n$ such that for any point $y \in N_x$ $X_1(y),\ldots,X_n(y)$ span $\Delta_y$ We let $\Delta$ refer to the collection of all the $\Delta_x$ for all $x \in M$ and we then call $\Delta$ a distribution of dimension $n$ on $M$ or sometimes a $C^\infty$ $n$ plane distribution on $M$ The set of smooth vector fields $\{ X_1,\ldots,X_n \}$ is called a local basis of $\Delta$
Note: The naming is unfortunate here as these distributions have nothing to do with distributions in the sense of analysis. However the naming is in wide use.
Definition 2 We say that a distribution $\Delta$ on $M$ is involutive if for every point $x \in M$ there exists a local basis $\{ X_1,\ldots,X_n \}$ in a neighbourhood of $x$ such that for all $1 \leq i, j \leq n$ $[X_i,X_j]$ (the commutator of two vector fields) is in the span of $\{ X_1,\ldots,X_n \}$ That is, if $[X_i,X_j]$ is a linear combination of $\{ X_1,\ldots,X_n \}$ Normally this is written as $[ \Delta , \Delta ] \subset
\Delta$
- 1
- William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.
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"distribution" is owned by jirka.
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See Also: Frobenius' theorem
| Other names: |
C^\infty n-plane distribution |
| Also defines: |
involutive, involutive distribution, local basis |
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Cross-references: linear combination, commutator, collection, span, point, vector fields, linearly independent, neighbourhood, tangent space, subspace, dimension, smooth manifold, smooth
There are 27 references to this entry.
This is version 3 of distribution, born on 2004-11-30, modified 2005-03-07.
Object id is 6541, canonical name is Distribution5.
Accessed 9025 times total.
Classification:
| AMS MSC: | 53-00 (Differential geometry :: General reference works ) |
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Pending Errata and Addenda
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