# The Unapologetic Mathematician

## Integral Submanifolds

Given a $k$-dimensional distribution $\Delta$ on an $n$-dimensional manifold $M$, we say that a $k$-dimensional submanifold $\iota:N\hookrightarrow M$ is an “integral submanifold” of $\Delta$ if $\iota_*\mathcal{T}_pN=\Delta_{\iota(p)}$ for every $p\in N$. That is, if the subspace of $\mathcal{T}_{\iota(p)}M$ spanned by the images of vectors from $\mathcal{T}_pN$ is exactly $\Delta_p$.

This is a lot like an integral curve, with one slight distinction: in the case on an integral curve we also demand that the length of $c'(t)$ match that of $X_{c(t)}$, not just the direction (up to sign).

Now, if for every $p\in M$ there exists an integral submanifold $N(p)$ of $\Delta$ with $p\in N(p)$, then $\Delta$ is integrable. Indeed, let $X$ and $Y$ belong to $\Delta$. Since $\iota_{*q}:N(p)_q\to\Delta_{\iota(q)}$ is an isomorphism of vector spaces at every point, we can find $\tilde{X}$ and $\tilde{Y}$ that are $\iota$-related to $X$ and $Y$, respectively. That is, $X_{\iota(q)}=\iota_*\tilde{X}_q$ for all $q\in N$, and similarly for $Y$ and $\tilde{Y}$. But then we know that $[X,Y]_{\iota(q)}=\iota_*[\tilde{X},\tilde{Y}]_q$, and so $[X,Y]_{\iota(q)}\in\iota_*\mathcal{T}_qN=\Delta_{\iota(q)}$.

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Posted by John Armstrong | Differential Topology, Topology

## 2 Comments	»

1. [...] around such that on . Then the curve with coordinates and for all other is a one-dimensional integral submanifold of through [...]

Pingback by | June 30, 2011 | Reply

2. [...] a -dimensional distribution on , which we say is “induced” by , and that any connected integral submanifold of should be contained in a leaf of . It makes sense, then, that we should call a leaf of a [...]

Pingback by | July 1, 2011 | Reply

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## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
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