Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Definition 9.1. A subset $S$ of a manifold $N$ of dimension $n$ is a regular submanifold of dimension $k$ if for every $p\in S$ there is a coordinate neighborhood $(U,\phi) = (U,x^{1},...,x^{n})$ of $p$ in the maximal atlas of $N$ such that $U\cap S$ is defined by the vanishing of $n-k$ coordinate functions.

What does he mean by "$U\cap S$ is defined by...". To me it seems the definition should be that $n-k$ coordinate functions vanish on $U\cap S$ and that's it. I understand the analogy with the $xy-plane$ being defined by the vanishing of the $z$ coordinate function in $R^{3}$, but am not seeing how the same terminology can be applied to the general manifold.

share|cite|improve this question
Your interpretation is precisely what Tu means. – Ryan Budney May 1 '11 at 18:56
up vote 7 down vote accepted

To rephrase the definition, for each $p\in S$, we have a neighborhood $U$ of $p$ and a coordinate chart $U\to V\subset \mathbb{R}^n$ of $N$ such that $S\cap U$ is the inverse image of a the intersection of $V$ with a $k$-dimensional linear subspace of $\mathbb{R}^n$ (in particular, a subspace of the form $(\star,\star,\star,\cdots, 0,0,0)$).

Locally, things do look like the $xy$-plane embedded in $\mathbb{R}^3$, defined by the vanishing of the $z$ coordinate. Of course, this won't hold for every chart, but we can cover $S$ with charts of $N$ such that, locally, a chart will determine what the points of $S$ are. If you have a point, and you have a chart, that tells you what the other points of $S$ are near $p$.

By the implicit function theorem, if $S$ is any submanifold of $N$, and $p\in S$, we can find a coordinate chart such that near $p$ the inclusion of $S$ into $N$ looks like the inclusion if $\mathbb{R}^k\subset \mathbb{R}^n$. Phrased differently, the condition that you wanted to hold happens for EVERY submanifold. So what this definition does is gives a topological restriction that you can't have different parts of the submanifold coming too close together.

For example, if you consider a slight modification of the topologist's sine curve, where you include a segment $(-1,1)$ of the $y$-axis and we connect that up with a path to the right hand side of the curve, you can view it as the image of the interval $(0,1)$ into $\mathbb{R}^2$ under a smooth, injective map, and so it is a submanifold of sorts. However, it is not a regular submanifold. Indeed, the induced topology (as a subset of $\mathbb{R}^2$ is different than the usual topology on $(-1,1)$). This is what the definition is meant to prevent, I believe.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.