Embeddings
November 5, 2019
1 Introduction
In advanced calculus, we have studied inverse function theorem (or equivalently implicit function theorem). They tell us, for one thing, that a function at a point behaves like its derivative. The derivative, as a linear map between vector spaces, is essentially determined by its rank (dimension of the image) up to choice of basis. From this point of view, the rank of the derivative determines the map up to diffeomorphism.
Since manifolds are locally Euclidean, it is natural to expect that some forms of those results continue to hold true. We will study maps whose differentials have constant rank and explore more properties in the next note.
2 Maps of Constant Rank
The maps of particular interests are ones whose differentials have constant rank throughout the manifold.
Definition 2.1 (Maps of Constant Rank). A smooth map F : M → N is of constant rank r if the linear map dFp : TpM → TF (p)N is of rank r at every point p ∈ M .
Definition 2.3 (dimM ≥ dimN ).
Let’s observe some examples:
It is not hard to see that being full rank is an open condition:
Proposition 2.4. Suppose F : M → N smooth and dFp is full ranked, then there exists open subset U around p so that dF is full ranked on U .
A classical (and very important) theorem in advanced calculus about local behavior of a smooth function versus its derivative is the inverse function the- orem. It says that if a map has nonsingular (i.e.e invertible) derivative at a point, then near that point the function is invertible with smooth inverse. In other words, it is a local diffeomorphism.
Definition 2.5. A map F : M → N is a local diffeomorphism if there exists a neighborhood U of p ∈ M so that F |U : U → F (U ) is a diffeomorphism, for every p ∈ M .
We have
Theorem 2.6 (Inverse Function Theorem). Given a smooth map F : M → N and p ∈ M so that dFp is invertible. Then there exists neighborhood U0, V0of p and F (p), respectively, such that F |U0: U0→ V0 is a diffeomorphism.
We can prove
Theorem 2.8. Inverse function theorem is equivalent to the rank theorem.
Rank theorem implies the following global consequence:
Theorem 2.9. Suppose F : M → N is a smooth map of constant rank, then
• If F is surjective, then it is a submersion.
• If F is injective, then it is an immersion.
• If F is bijective, then it is a diffeomorphism.
Injective smooth immersion is not exactly a smooth embedding, but a local one.
Proposition 2.10. Given an injective smooth immersion F : M → N , it is a smooth embedding if any of the followings is true:
a F is open or closed.
b M is compact.
c F is proper (i.e. F−1(K) is compact for K ⊂ N compact.)
Theorem 2.11. Given smooth map F : M → N , then it is an smooth immer- sion if and only if every point of M has a neighborhood on which F is a smooth embedding.
3 Submersion
Submersions between smooth manifolds resemble quotient maps between topo- logical spaces. Recall
Definition 3.1. A map between topological space π : X → Y is called a quo- tient map if it is surjective, continuous, and Y has quotient topology determined by π.
An important object associated with submersion is the notion of section:
Definition 3.2. Given continuous map π : M → N , a section of π is a map σ : N → M so that π ◦ σ = idN. If σ is only defined locally, it is called a local section
The rank theorem implies the following theorem to characterize submersions:
Theorem 3.3. Given smooth map F : M → N , then F is a smooth submersion if and only if every point of M is in the image of a smooth local section of F .
The previous theorem immediately gives rise to the following example of submersion:
Example 3.4. The projection map π : T M → M is a submersion.
With the theorem, we see that submersion is almost a quotient map:
Theorem 3.5. Given submersion π : M → N , then π is an open map. It is a quotient map if it is surjective.
Similar to quotient maps, smooth maps o N are in correspondence with their lifts to M :
Theorem 3.6 (Characteristic Property of Surjective Smooth Submersons). Let π : M → N be smooth surjective submersion, and P another manifold. Then a
On the other hand, a smooth map F : M → P can always be ”pushed down”
to a map on its quotient, as long as it respects each fiber π−1(p):
Theorem 3.7. Let π : M → N be a smooth surjective submersion, and F : M → P is a smooth map, then there exists a unique map ˜F : N → P such that F ◦ π = F .˜
