nLab
submersion

Contents

Context

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

  • (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 }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Étale morphisms

          Contents

          Definition

          Let XX and YY be two smooth manifolds of finite dimension and let f:XYf : X \to Y be a differentiable function between them

          In components, the definition of submersion reads as follows.

          Definition

          The function f:XYf : X \to Y is called a submersion precisely if its differential df:TXTYd f\colon T X \to T Y is for every point xXx \in X a surjection df x:T xXT f(x)Yd f_x\colon T_x X \to T_{f(x)} Y.

          More abstractly formulated, this means equivalently the following.

          Definition

          The function f:XYf : X \to Y is a submersion precisely if the canonical morphism

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

          from the tangent bundle of XX to the pullback of the tangent bundle of YY along ff is a surjection.

          This morphism is the one induced by the universal property of the pullback from 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 } \,.

          In terms of coordinates, the map ff is a submersion at a point p:Xp\colon X if and only if there exists a coordinate chart on XX near pp and a coordinate chart on YY near f(p)f(p) relative to which ff is the projection f(x 1,,x n)=(x 1,,x m)f(x_1,\ldots,x_n) = (x_1,\ldots,x_m). This definition applies to infinite-dimensional manifolds, to non-differentiable maps, even between non-differentiable manifolds.

          Properties

          Pullbacks

          While the category Diff of (finite dimensional) smooth manifolds does not have all pullbacks, the pullback along a submersion always exists. This is because a submersion is transversal to every other smooth map into its codomain. Moreover, submersions are stable under pullback.

          Epimorphisms and coverings

          The surjective submersions (that is the submersions that are also epimorphisms in Diff) are regular epimorphisms.

          Surjective submersions form a singleton Grothendieck pretopology on Diff, and so may be used in internal category theory when using DiffDiff as the ambient category. They appear notably in the definition of Lie groupoids.

          Ehresmann's theorem states that a proper submersion is a locally trivial fibration.

          Normal form

          For f:XYf : X \to Y a submersion, then around every point of XX there is an open neighbourhood on which ff restricts to a projection.

          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 smooth morphism in SynthDiffGrpdSynthDiff\infty Grpd.

          ff is a submersion precisely if it is formally smooth 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.

          References

          For instance chapter XIV

          • Serge Lang, Fundamentals of differential geometry Springer (1991)

          Ehresmann’s theorem is due to

          • Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Bruxelles (1950), 29-55.

          Last revised on October 9, 2017 at 19:37:57. See the history of this page for a list of all contributions to it.