nLab
immersion of smooth manifolds

Definition

Let $X$ and $X$ be smooth manifolds of finite dimension. Let $f:X\to Y$ be a differentiable function.

Definition

The function $f:X\to Y$ is called an immersion precisely if the canonical morphism

$$TX\to X{\times }_{Y}TY≕f^{*}TY$$T X \to X \times_Y T Y \eqqcolon f^* T Y

is a monomorphism.

This morphism is the one induced from the universal property of the pullback by the commuting diagram

$$\begin{array}{ccc}TX& \stackrel{df}{\to }& TY\\ \downarrow & & \downarrow \\ X& \stackrel{f}{\to }& Y\end{array}$$\array{ T X &\stackrel{d f}{\to}& T Y \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y }

given by the differential of $f$ going between the tangent bundles.

Equivalently this means the following.

Definition

The function $f:X\to Y$ is an immersion precisely if for every point $x\in X$ the differential

$$df{\mid }_x:T_{x}X\to T_{f(y)}Y$$d f|_x : T_x X \to T_{f(y)} Y

between the tangent space of $X$ at $x$ and the tangent space of $Y$ at $f(y)$ is an injection.

An immersion whose underlying continuous function is an embedding of topological spaces is an embedding of smooth manifolds.

Properties

Relation to embeddings

An immersion $f:X\to Y$ is precisely a local embeddings: for every point $x\in X$ there is an open neighbourhood $x\in U\subset X$ such that $f{\mid }_U:U\to Y$ is an embedding of smooth manifolds.

Characterization in infinitesimal cohesion

A smooth function $f:X\to Y$ between smooth manifolds is canonically regarded as a morphism in the cohesive (∞,1)-topos SynthDiff∞Grpd. With respect to the canonical infinitesimal neighbourhood inclusion $i:$ Smooth∞Grpd $\hookrightarrow$ SynthDiff∞Grpd there is a notion of formally unramified morphism in $\mathrm{SynthDiff}\mathrm{\infty }\mathrm{Grpd}$.

$f$ is an immersion precisely if it is formally unramified with respect to this infinitesimal cohesion.

See the discussion at SynthDiff∞Grpd for details.

Variants

The algebraic geometry analogue of a submersion is a smooth morphism.

The analogue between arbitrary topological spaces (not manifolds) is simply an open map. There is also topological submersion, of which there are two versions.

Related concepts

• immersion, submersion, local diffeomorphism

• embedding of smooth manifolds