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




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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Étale morphisms



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.


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.


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

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

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.



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.


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.


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.

Revised on May 16, 2017 13:19:55 by Urs Schreiber (