nLab immersion of smooth manifolds



Étale morphisms

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



Let f:XYf \colon X \to Y be a differentiable function between smooth manifolds (or just differentiable manifolds) of finite dimension.

We denote by TXT X, TYT Y the tangent bundles and by ()× Y()(-)\times_Y (-) the fiber product of differentiable functions into YY. In particular, f *TY=X× YTYf^\ast T Y \,=\, X \times_Y T Y is the pullback bundle of TYT Y along ff to a real vector bundle over XX.


The differentiable function f:XYf : X \to Y is called an immersion precisely if the canonical map

TXX× YTYf *TY T X \longrightarrow X \times_Y T Y \eqqcolon f^* T Y

is a monomorphism.

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

TX df TY X f Y \array{ T X &\overset{d f}{\longrightarrow}& T Y \\ \big\downarrow && \big\downarrow \\ X &\underset{f}{\longrightarrow}& Y }

given by the differential of ff going between the tangent bundles.

Equivalently this means the following:


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

df| x:T xXT f(x)Y d f|_x \;\colon\; T_x X \to T_{f(x)} Y

between the tangent space of XX at xx and the tangent space of YY at f(y)f(y) is an injection.


(immersions that are not embeddings)

Consider an immersion f:(a,b) 2f \;\colon\; (a,b) \to \mathbb{R}^2 of an open interval into the Euclidean plane (or the 2-sphere) as shown on the right. This is not an embedding of smooth manifolds: around the points where the image crosses itself, the function is not even injective, but even at the points where it just touches itself, the pre-images under ff of open subsets of 2\mathbb{R}^2 do not exhaust the open subsets of (a,b)(a,b), hence do not yield the subspace topology.

Concretely, consider the function (\big(sin(2),(2-), sin()):(π,π) 2(-)\big) \;\colon\; (-\pi, \pi) \longrightarrow \mathbb{R}^2. While this is an immersion and injective, it fails to be an embedding due to the points at t=±πt = \pm \pi “touching” the point at t=0t = 0.

figure from Lee (2012, Fig. 4.3)


Relation to embeddings

Immersion are precisely the “local embeddings”:


A smooth map f:XYf \colon X \to Y of smooth manifolds is an immersion precisely if for every point xXx \in X there is an open neighbourhood xUXx \in U \subset X such that f| U:UYf|_U \colon U \to Y is an embedding of smooth manifolds.

(e.g. Boothby, III Thm. 4.12)

Since moreover the embeddings of smooth manifolds admit local slice charts (see there) so do immersions.

In the generality of supermanifolds this is Varadarajan 2004, Thm. 4.4.3.

Relation to formal immersions

The related concept of formal immersion of smooth manifolds, defined as an injective bundle morphism TMTNT M \to T N between tangent bundles, is in some ways easier to study, in the sense that the collection of all such formal immersions, Imm f(M,N)Imm^f(M, N), is simpler to analyze. Then under some conditions on MM and NN (see there), it is the case that the map Imm(M,N)Imm f(M,N)Imm(M, N) \to \Imm^f(M, N) is a weak homotopy equivalence. This is a case of the h-principle.

Characterization in differential cohesion

A smooth function f:XYf : 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:i : Smooth∞Grpd \hookrightarrow SynthDiff∞Grpd there is a notion of formally unramified morphism in SynthDiffGrpdSynthDiff\infty Grpd.

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

See the discussion at SynthDiff∞Grpd for details.


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.


Discussion in the generality of supermanifolds:

Last revised on May 18, 2024 at 14:28:13. See the history of this page for a list of all contributions to it.