MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 added 1 characters in body

I like to begin an introduction to manifolds by proving that the following conditions on a subset of $\mathbb R^n$ are equivalent:

1. Locally it can be mapped to an open subset of $\mathbb R^p\times 0$ by a diffeomorphism between open subsets of $\mathbb R^n$.

2. Locally it appears as the graph of a smooth function expressing $n-p$ of the coordinates as functions of the other $p$.

3. Locally it is $f^{-1}(0)$ for a smooth map to $\mathbb R^{n-p}$ whose derivative has maximal rank (i.e. rank $n-p$).

4. Locally it is the image of a smooth map from $\mathbb R^p$ of maximal rank (i.e. rank $p$).

2 implies 1 easily. 1 implies 3 and 4 easily. 4 implies 2 by the inverse function theorem in dimension $p$. 3 implies 2 (this implication might be called the implicit function theorem) by the inverse function theorem in dimension $n$.

I like to think that any one of these four would serve as a good definition of "smooth $p$-dimensional manifold in $\mathbb R^n$" for those who have not studied abstract manifolds, but that you haven't really begun to get into the subject until you see that they are the same.

1

I like to begin an introduction to manifolds by proving that the following conditions on a subset of $\mathbb R^n$ are equivalent:

1. Locally it can be mapped to an open subset of $\mathbb R^p\times 0$ by a diffeomorphism between open subsets of $\mathbb R^n$.

2. Locally it appears as the graph of a smooth function expressing $n-p$ of the coordinates as functions of the other $p$.

3. Locally it is $f^{-1}(0)$ for a smooth map to $\mathbb R^{n-p}$ whose derivative has maximal rank (i.e. rank $n-p$).

4. Locally it is the image of a smooth map from $\mathbb R^p$ of maximal rank (i.e. rank $p$).

2 implies 1 easily. 1 implies 3 and 4 easily. 4 implies 2 by the inverse function theorem in dimension $p$. 3 implies 2 (this implication might be called the implicit function theorem) by the inverse function theorem in dimension $n$.

I like to think that any one of these four would serve as a good definition of "smooth $p$-dimensional manifold in $\mathbb R^n$ for those who have not studied abstract manifolds, but that you haven't really begun to get into the subject until you see that they are the same.