Smooth immersion - Imported Knowlpedia Content

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

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

d f_p : T_p M \longrightarrow T_{f(p)}N

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

Equivalent characterizations

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

Examples

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 $\mathbb{R}^m \hookrightarrow \mathbb{R}^n$ .

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 $S^1\hookrightarrow \mathbb{R}^2$ .

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.

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:

(\psi\circ f \circ \varphi^{-1})(x^1,\dots,x^m)=(x^1,\dots,x^m,0,\dots,0)\in \mathbb{R}^n.

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.