integral manifold

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

Definition.

Let $M$ be a smooth manifold of dimension $m$ and let $\mathrm{\Delta }$ 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 ${\mathrm{\Delta }}_x$ (the distribution at $x$). We can abbreviate this by saying that $T(N)\subset \mathrm{\Delta }$. We then say that $N$ is an integral manifold of $\mathrm{\Delta }$.

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

Definition.

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

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.

Title	integral manifold
Canonical name	IntegralManifold
Date of creation	2013-03-22 14:52:00
Last modified on	2013-03-22 14:52:00
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	6
Author	jirka (4157)
Entry type	Definition
Classification	msc 53B25
Classification	msc 52-00
Classification	msc 37C10
Related topic	FrobeniussTheorem
Defines	completely integrable
Defines	completely integrable distribution