Submersion (mathematics)
In
Definition
Let M and N be differentiable manifolds and be a
is a
A word of warning: some authors use the term critical point to describe a point where the
Submersion theorem
Given a submersion between smooth manifolds of dimensions and , for each there are surjective charts of around , and of around , such that restricts to a submersion which, when expressed in coordinates as , becomes an ordinary orthogonal projection. As an application, for each the corresponding fiber of , denoted can be equipped with the structure of a smooth submanifold of whose dimension is equal to the difference of the dimensions of and .
The theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure).
For example, consider given by The Jacobian matrix is
This has maximal rank at every point except for . Also, the fibers
are empty for , and equal to a point when . Hence we only have a smooth submersion and the subsets are two-dimensional smooth manifolds for .
Examples
- Any projection
- Local diffeomorphisms
- Riemannian submersions
- The projection in a smooth local trivialization.
Maps between spheres
One large class of examples of submersions are submersions between spheres of higher dimension, such as
whose fibers have dimension . This is because the fibers (inverse images of elements ) are smooth manifolds of dimension . Then, if we take a path
and take the
we get an example of a special kind of bordism, called a framed bordism. In fact, the framed cobordism groups are intimately related to the
Families of algebraic varieties
Another large class of submersions are given by families of algebraic varieties whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstrass family of elliptic curves is a widely studied submersion because it includes many technical complexities used to demonstrate more complex theory, such as intersection homology and perverse sheaves. This family is given by
where is the affine line and is the affine plane. Since we are considering complex varieties, these are equivalently the spaces of the complex line and the complex plane. Note that we should actually remove the points because there are singularities (since there is a double root).
Local normal form
If f: M → N is a submersion at p and f(p) = q ∈ N, then there exists an
It follows that the full preimage f−1(q) in M of a regular value q in N under a differentiable map f: M → N is either empty or is a differentiable manifold of dimension dim M − dim N, possibly disconnected. This is the content of the regular value theorem (also known as the submersion theorem). In particular, the conclusion holds for all q in N if the map f is a submersion.
Topological manifold submersions
Submersions are also well-defined for general topological manifolds.[3] A topological manifold submersion is a continuous surjection f : M → N such that for all p in M, for some continuous charts ψ at p and φ at f(p), the map ψ−1 ∘ f ∘ φ is equal to the projection map from Rm to Rn, where m = dim(M) ≥ n = dim(N).
See also
- Ehresmann's fibration theorem
Notes
- ^ Crampin & Pirani 1994, p. 243. do Carmo 1994, p. 185. Frankel 1997, p. 181. Gallot, Hulin & Lafontaine 2004, p. 12. Kosinski 2007, p. 27. Lang 1999, p. 27. Sternberg 2012, p. 378.
- ^ Arnold, Gusein-Zade & Varchenko 1985.
- ^ Lang 1999, p. 27.
References
- ISBN 0-8176-3187-9.
- Bruce, James W.; Giblin, Peter J. (1984). Curves and Singularities. MR 0774048.
- Crampin, Michael; Pirani, Felix Arnold Edward (1994). Applicable differential geometry. Cambridge, England: ISBN 978-0-521-23190-9.
- ISBN 978-0-8176-3490-2.
- Frankel, Theodore (1997). The Geometry of Physics. Cambridge: MR 1481707.
- Gallot, Sylvestre; ISBN 978-3-540-20493-0.
- Kosinski, Antoni Albert (2007) [1993]. Differential manifolds. Mineola, New York: Dover Publications. ISBN 978-0-486-46244-8.
- ISBN 978-0-387-98593-0.
- ISBN 978-0-486-47855-5.