nLab
immersion of smooth manifolds

Context

Étale morphisms

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential 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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

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

Definition

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

TXX× YTYf *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

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

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

Equivalently this means the following:

Definition

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

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

Example

(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 a 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.

As a concrete examples, consider the function (sin(2),sin()):(π,π) 2(sin(2-), sin(-)) \;\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.

graphics grabbed from Lee

Properties

Relation to embeddings

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

Characterization in infinitesimal 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.

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.

Revised on May 16, 2017 13:16:38 by Urs Schreiber (131.220.184.222)