Smooth immersion

Let $M$ and $N$ be smooth manifolds of dimensions $m$ and $n$, and let $f : M \to N$ be a smooth map .

Definition

The map $f$ is a smooth immersion if for every $p \in M$, the differential (pushforward)

is injective, where $T_p M$ denotes the tangent space at \(p\) .

Equivalently, $\operatorname{rank}(d f_p)=m$ for all $p\in M$. In particular, if an immersion $M^m \to N^n$ exists then $m \le n$.

Local normal form (Immersion theorem)

If $f : M^m \to N^n$ is a smooth immersion and $p\in M$, then there exist smooth charts $(U,\varphi)$ about $p$ and $(V,\psi)$ about $f(p)$ (see coordinate charts ) such that, in coordinates, $f$ becomes the standard inclusion:

Thus an immersion is locally an embedding into $N$, but globally it need not be injective. When an immersion is also a homeomorphism onto its image (with the subspace topology), it is a smooth embedding . A diffeomorphism is in particular both a smooth immersion and a smooth submersion .

Examples

1. Linear inclusion. The map $i:\mathbb{R}^m \to \mathbb{R}^n$ for $m\le n$ defined by

   $$i(x^1,\dots,x^m)=(x^1,\dots,x^m,0,\dots,0)$$

   is a smooth immersion: for every point, the differential is the injective inclusion $\mathbb{R}^m \hookrightarrow \mathbb{R}^n$.

2. Standard circle in the plane. The map $f:\mathbb{R}\to \mathbb{R}^2$ given by

   $$f(t)=(\cos t,\sin t)$$

   is a smooth immersion since $f'(t)=(-\sin t,\cos t)\neq 0$ for all $t$. It is not injective as a map $\mathbb{R}\to\mathbb{R}^2$, but it factors through the quotient $\mathbb{R}\to S^1$ to give an embedding $S^1\hookrightarrow \mathbb{R}^2$.

3. An immersion with self-intersections. The “figure-eight” map $g:S^1\to \mathbb{R}^2$ defined (using an angle parameter $t$) by

   $$g(t)=(\sin t,\sin 2t)$$

   is a smooth immersion: $g'(t)=(\cos t,2\cos 2t)$ never vanishes. However, $g(0)=g(\pi)=(0,0)$, so it is not a smooth embedding .