= Solution
An <immersed submanifold> of $M$ is a manifold $Y$ equipped with an injective <immersion> $\psi:Y\to M$. It is an <embedded submanifold> when $\psi$ is also a homeomorphism onto its image with the subspace topology, equivalently when it is a <smooth embedding>. For irrational $\alpha$, the <irrational winding of the torus>
$$
t\longmapsto(e^{it},e^{i\alpha t})
$$
is an injective immersion $\mathbb R\to T^2$ with dense image, and therefore is not an embedding. If $Y$ is compact, however, an injective immersion is a continuous bijection from a <compact> space to its image in the <Hausdorff> manifold $M$; its inverse is continuous. Thus the <compact injective immersion is an embedding> theorem makes $\psi(Y)$ embedded.
Now let $X\subset M$ be embedded, with $\dim M=m$, $\dim X=m-k$, and $p\in X$. Apply the <constant rank theorem> to its inclusion. After choosing coordinates and reordering them, there is a neighborhood $U$ of $p$ with coordinates $(x^1,\ldots,x^m)$ for which
$$
\boxed{U\cap X=\{x^1=\cdots=x^k=0\}.}
$$
This is the <slice chart for an embedded submanifold>.
It is \b[false] that every embedded submanifold is the inverse image of a regular value of a map to a <Euclidean space>. If a codimension-$k$ submanifold is $f^{-1}(y)$ for a <regular value> of $f:M\to\mathbb R^k$, the differentials of the component functions give a global frame of its conormal bundle, so its <normal bundle> is trivial. The core circle of the <Möbius band> is embedded but has the nontrivial Möbius normal line bundle. This is the <normal-bundle obstruction to being a regular level set>.
Finally, an inductive spinning construction gives the requested torus. Place an embedding $F:N^m\hookrightarrow\mathbb R^{m+1}$ in the half-space $F_{m+1}>0$ and define
$$
\widetilde F(x,e^{i\theta})=
\bigl(F_1(x),\ldots,F_m(x),F_{m+1}(x)\cos\theta,F_{m+1}(x)\sin\theta\bigr).
$$
The positive radius makes this map injective, and its differential is injective in both the $N$ and circle directions; compactness then makes it an embedding. Starting with $S^1\hookrightarrow\mathbb R^2$ and iterating proves the <embedding of the n-dimensional torus in codimension one>:
$$
\boxed{T^n\hookrightarrow\mathbb R^{n+1}.}
$$
Back to article page
