For other meanings, see Distribution (disambiguation).

In differential geometry, a discipline within mathematics, a distribution is a subset of the tangent bundle of a manifold satisfying certain properties. Distributions are used to build up notions of integrability, and specifically of a foliation of a manifold.

Definition

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Let $M$ be a $C^\backslash infty$ manifold of dimension $m$, and let $n\; \backslash leq\; m$. Suppose that for each $x\; \backslash in\; M$, we assign an $n$-dimensional subspace $\backslash Delta\_x\; \backslash subset\; T\_x(M)$ of the tangent space in such a way that for a neighbourhood $N\_x\; \backslash subset\; M$ of $x$ there exist $n$ linearly independent smooth vector fields $X\_1,\backslash ldots,X\_n$ such that for any point $y\; \backslash in\; N\_x$, $X\_1(y),\backslash ldots,X\_n(y)$ span $\backslash Delta\_y.$ We let $\backslash Delta$ refer to the collection of all the $\backslash Delta\_x$ for all $x\; \backslash in\; M$ and we then call $\backslash Delta$ a distribution of dimension $n$ on $M$, or sometimes a $C^\backslash infty$ $n$-plane distribution on $M.$ The set of smooth vector fields $\backslash \{\; X\_1,\backslash ldots,X\_n\; \backslash \}$ is called a local basis of $\backslash Delta.$

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.

Involutive distributions

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We say that a distribution $\backslash Delta$ on $M$ is involutive if for every point $x\; \backslash in\; M$ there exists a local basis $\backslash \{\; X\_1,\backslash ldots,X\_n\; \backslash \}$ of the distribution in a neighbourhood of $x$ such that for all $1\; \backslash leq\; i,\; j\; \backslash leq\; n$, is a linear combination of $\backslash \{\; X\_1,\backslash ldots,X\_n\; \backslash \}.$ Normally this is written as $\backslash Delta\; ,\; \backslash Delta\backslash subset\; \backslash Delta.$

Involutive distributions are the tangent spaces to foliations. Involutive distributions are important in that they satisfy the conditions of the Frobenius theorem, and thus lead to integrable systems.

A related idea occurs in Hamiltonian mechanics: two functions f and g on a symplectic manifold are said to be in mutual involution if their Poisson bracket vanishes.

References

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• William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.

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Differential geometry | Foliations