In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential pushforward is everywhere surjective. It is a basic concept in differential topology, dual to that of an immersion.
Contents
Definition
Let M and N be differentiable manifolds, and let
f
:
M
→
N
{\displaystyle f\colon M\to N}
be a differentiable map between them. The map f is a submersion at a point
p
∈
M
{\displaystyle p\in M}
if its differential
D
f
p
:
T
p
M
→
T
f
(
p
)
N
{\displaystyle Df_{p}\colon T_{p}M\to T_{f(p)}N}
is a surjective linear map. In this case, p is called a regular point of the map f; otherwise, p is a critical point. A point
q
∈
N
{\displaystyle q\in N}
is a regular value of f if all points p in the preimage
f
−
1
(
q
)
{\displaystyle f^{-1}(q)}
are regular points. A differentiable map f that is a submersion at each point
p
∈
M
{\displaystyle p\in M}
is called a submersion. Equivalently, f is a submersion if its differential
D
f
p
{\displaystyle Df_{p}}
has constant rank equal to the dimension of N.
Some authors use the term critical point to describe a point where the rank of the Jacobian matrix of f at p is not maximal.: Indeed, this is the more useful notion in singularity theory. If the dimension of M is greater than or equal to the dimension of N, then these two notions of critical point coincide. However, if the dimension of M is less than the dimension of N, all points are critical according to the definition above (the differential cannot be surjective), but the rank of the Jacobian may still be maximal (if it is equal to dim M). The definition given above is the more commonly used one, e.g., in the formulation of Sard's theorem.
Submersion theorem
Given a submersion
f
:
M
→
N
{\displaystyle f\colon M\to N}
between smooth manifolds of dimensions
m
{\displaystyle m}
and
n
{\displaystyle n}
, for each
x
∈
M
{\displaystyle x\in M}
there exist surjective charts
ϕ
:
U
→
R
m
{\displaystyle \phi :U\to \mathbb {R} ^{m}}
of
M
{\displaystyle M}
around
x
{\displaystyle x}
Examples
Any projection
π
:
R
m
+
n
→
R
n
⊂
R
m
+
n
{\displaystyle \pi \colon \mathbb {R} ^{m+n}\rightarrow \mathbb {R} ^{n}\subset \mathbb {R} ^{m+n}}
Local diffeomorphisms
Riemannian submersions
The projection in a smooth vector bundle or a more general smooth fibration. The surjectivity of the differential is a necessary condition for the existence of a local trivialization.
Maps between spheres
A large class of examples of submersions are submersions between spheres of higher dimension, such as
f
:
S
n
+
k
→
S
k
{\displaystyle f:S^{n+k}\to S^{k}}
whose fibers have dimension
n
{\displaystyle n}
. This is because the fibers (inverse images of elements
p
∈
S
k
{\displaystyle p\in S^{k}}
) are smooth manifolds of dimension
n
{\displaystyle n}
. Then, if we take a path
γ
:
I
→
S
k
{\displaystyle \gamma :I\to S^{k}}
Families of algebraic varieties
Another large class of submersions is given by families of algebraic varieties
π
:
X
→
S
{\displaystyle \pi :{\mathfrak {X}}\to S}
whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstrass family
π
:
W
→
A
1
{\displaystyle \pi :{\mathcal {W}}\to \mathbb {A} ^{1}}
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
W
=
{
(
t
,
x
,
y
)
∈
A
Local normal form
If f: M → N is a submersion at p and f(p) = q ∈ N, then there exists an open neighborhood U of p in M, an open neighborhood V of q in N, and local coordinates (x1, …, xm) at p and (x1, …, xn) at q such that f(U) = V, and the map f in these local coordinates is the standard projection
f
(
x
1
,
…
,
x
n
,
x
n
+
1
,
…
,
x
m
)
=
(
x
1
,
…
,
x
n
)
.
{\displaystyle f(x_{1},\ldots ,x_{n},x_{n+1},\ldots ,x_{m})=(x_{1},\ldots ,x_{n}).}
Topological manifold submersions
Submersions are also well-defined for general topological manifolds. 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).