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Homedistribution

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# distribution

###### Definition.

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.

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 $[\Delta,\Delta]\subset\Delta$.

# References

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

## Mathematics Subject Classification

53-00*no label found*

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