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




  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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 }



          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)



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


          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:


          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.


          (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


          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.

          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.

          Last revised on July 25, 2018 at 12:10:48. See the history of this page for a list of all contributions to it.