# integral manifold

In the following we will $C^{\infty}$ when we say smooth.

###### Definition.

Let $M$ be a smooth manifold  of dimension $m$ and let $\Delta$ be a distribution of dimension $n$ on $M$. Suppose that $N$ is a connected submanifold  of $M$ such that for every $x\in N$ we have that $T_{x}(N)$ (the tangent space of $N$ at $x$) is contained in $\Delta_{x}$ (the distribution at $x$). We can abbreviate this by saying that $T(N)\subset\Delta$. We then say that $N$ is an integral manifold of $\Delta$.

Do note that $N$ could be of lower dimension then $\Delta$ and is not required to be a regular submanifold of $M$.

###### Definition.

We say that a distribution $\Delta$ of dimension $n$ on $M$ is completely integrable if for each point $x\in M$ there exists an integral manifold $N$ of $\Delta$ passing through $x$ such that the dimension of $N$ is equal to the dimension of $\Delta$.

An example of an integral manifold is the integral curve of a non-vanishing vector field and then of course the span of the vector field is a completely integrable distribution.

## References

• 1 William M. Boothby. , Academic Press, San Diego, California, 2003.
Title integral manifold IntegralManifold 2013-03-22 14:52:00 2013-03-22 14:52:00 jirka (4157) jirka (4157) 6 jirka (4157) Definition msc 53B25 msc 52-00 msc 37C10 FrobeniussTheorem completely integrable completely integrable distribution