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# Frobenius’ theorem

###### Theorem (Frobenius).

Let $M$ be a smooth manifold ($C^{\infty}$) and let $\Delta$ be a distribution on $M$. Then $\Delta$ is completely integrable if and only if $\Delta$ is involutive.

One direction in the proof is pretty easy since the tangent space of an integral manifold is involutive, so sometimes the theorem is only stated in one direction, that is: If $\Delta$ is involutive then it is completely integrable.

Another way to state the theorem is that if we have $n$ vector fields $\{X_{k}\}_{{k=1}}^{n}$ on a manifold $M$ such that they are linearly independent at every point of the manifold, and furthermore if for any $k,m$ we have $[X_{k},X_{m}]=\sum_{{j=1}}^{n}a_{j}X_{j}$ for some $C^{\infty}$ functions $a_{j}$, then for any point $x\in N$, there exists a germ of a submanifold $N\subset M$, through $x$, such that $TN$ is spanned by $\{X_{k}\}_{{k=1}}^{n}$. Note that if we extend $N$ to all of $M$, it need not be an embedded submanifold anymore, but just an immersed one.

For $n=1$ above, this is just the existence and uniqueness of solution of ordinary differential equations.

# References

- 1 William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.
- 2 Frobenius theorem at Wikipedia: http://en.wikipedia.org/wiki/Frobenius_theorem

## Mathematics Subject Classification

37C10*no label found*53-00

*no label found*53B25

*no label found*

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