Let and be two smooth manifolds of finite dimension and let be a differentiable function between them
In components, the definition of submersion reads as follows.
The function is called a submersion precisely if its differential is for every point a surjection .
More abstractly formulated, this means equivalently the following.
The function is a submersion precisely if the canonical morphism
from the tangent bundle of to the pullback of the tangent bundle of along is a surjection.
This morphism is the one induced by the universal property of the pullback from the commuting diagram
In terms of coordinates, the map is a submersion at a point if and only if there exists a coordinate chart on near and a coordinate chart on near relative to which is the projection . 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 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.
For a submersion, then around every point of there is an open neighbourhood on which restricts to a projection.
Characterization in infinitesimal cohesion
A smooth function between smooth manifolds is canonically regarded as a morphism in the cohesive (∞,1)-topos SynthDiff∞Grpd. With respect to the canonical infinitesimal neighbourhood inclusion Smooth∞Grpd SynthDiff∞Grpd there is a notion of formally smooth morphism in .
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.