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 adistribution of dimension $n$ on $M$. Suppose that $N$ is a connectedsubmanifold 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 manifoldof $\mathrm{\Delta }$.

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 foreach 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 dimensionof $\mathrm{\Delta }$.

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