Smooth Embeddings
A
smooth immersion
is a map whose differential is injective at every point. The condition is purely local and purely
differential. The rank theorem tells us that an immersion looks locally like the standard
inclusion \((x^1, \dots, x^m) \mapsto (x^1, \dots, x^m, 0, \dots, 0)\), so its image is, in small
pieces, a well-behaved copy of the domain.
But globally an immersion can do unpleasant things. Its image may cross itself, accumulate on
itself, or carry a topology utterly unlike the domain's. To capture the notion of one manifold
sitting inside another as a faithful copy, we must add a topological condition to the differential
one. Faithful here means the same points, the same topology, and the same smooth structure.
Definition: Smooth Embedding
Let \(M\) and \(N\) be smooth manifolds, with or without boundary. A map \(F : M \to N\) is a
smooth embedding if it is a smooth immersion that is also a
topological embedding,
a homeomorphism onto its image \(F(M) \subseteq N\) endowed with the subspace topology.
The two requirements are independent, and both are essential. A smooth embedding is not
the same as a topological embedding that happens to be smooth. The differential condition can fail
even when the topological one holds. Nor is a smooth immersion automatically an embedding. The
topological condition can fail even when the map is injective. The interplay of these two
conditions is the entire content of this section and the next, and it is best understood through
examples, first the well-behaved ones, then three instructive failures.
The simplest embeddings are inclusions. If \(U \subseteq M\) is an open submanifold, the inclusion
\(U \hookrightarrow M\) is a smooth embedding. More generally, if \(M_1, \dots, M_k\) are smooth
manifolds and points \(p_i \in M_i\) are fixed for \(i \neq j\), the slice inclusion
\[
\iota_j(q) = (p_1, \dots, p_{j-1}, q, p_{j+1}, \dots, p_k)
\]
embeds \(M_j\) into the product \(M_1 \times \cdots \times M_k\). The prototype is the standard
inclusion \(\mathbb{R}^n \hookrightarrow \mathbb{R}^{n+k}\),
\((x^1, \dots, x^n) \mapsto (x^1, \dots, x^n, 0, \dots, 0)\). Compositions of smooth embeddings are
again smooth embeddings, so these basic examples generate many more.
Three ways an injective smooth map can fail to be an embedding
To see what the embedding condition genuinely demands, it is worth keeping in mind three injective
smooth maps that fall short of being embeddings, each in a different way.
Failure of the immersion condition.
The map
\(c : \mathbb{R} \to \mathbb{R}^2\), \(c(t) = (t^3, 0)\), is smooth and is a topological
embedding, a homeomorphism onto the \(x\)-axis. Yet it is not a smooth embedding, because
\(c'(0) = 0\), so it is not an immersion. This is the example to remember whenever one is tempted
to define embeddings as "smooth topological embeddings". The immersion requirement is doing real
work.
Failure through self-crossing.
Define
\(\beta : (-\pi, \pi) \to \mathbb{R}^2\) by \(\beta(t) = (\sin 2t, \sin t)\). Its image is a
figure-eight, or lemniscate, the locus \(x^2 = 4y^2(1 - y^2)\). One checks that
\(\beta'(t)\) never vanishes, so \(\beta\) is a smooth immersion, and that \(\beta\) is injective.
But it is not a topological embedding. As \(t \to \pi\) (and as \(t \to -\pi\)) the image point
\(\beta(t)\) returns toward the crossing point \(\beta(0) = (0,0)\). Thus both ends of the domain
accumulate at the single image point \(\beta(0)\). Every neighborhood of \(\beta(0)\) in the image
necessarily contains points \(\beta(t)\) with \(t\) near \(\pm\pi\), not only those with \(t\)
near \(0\). Pulling such a neighborhood back through \(\beta^{-1}\) therefore yields a set
containing parameters near \(\pm\pi\), which is not a neighborhood of \(0\) in \((-\pi,\pi)\).
Concretely, there is a sequence in the image converging to \(\beta(0)\) whose parameters run off
to the ends of \((-\pi, \pi)\) and do not converge in the domain. The corestriction
\(\beta : (-\pi,\pi) \to \beta((-\pi,\pi))\) is therefore a continuous bijection whose inverse is
discontinuous at the origin, hence not a homeomorphism.
Failure through accumulation.
Let
\(\mathbb{T}^2 = S^1 \times S^1 \subseteq \mathbb{C}^2\) be the torus, let \(\alpha\) be an
irrational number, and define \(\gamma : \mathbb{R} \to \mathbb{T}^2\) by
\(\gamma(t) = \big(e^{2\pi i t}, e^{2\pi i \alpha t}\big)\). Its velocity never vanishes, so it is
a smooth immersion. The map is injective too, because \(\gamma(t_1) = \gamma(t_2)\) would require
both \(t_1 - t_2\) and \(\alpha(t_1 - t_2)\) to be integers, impossible for irrational \(\alpha\)
unless \(t_1 = t_2\). Nevertheless \(\gamma\) is not a topological embedding. Using the
approximation lemma below, one shows that \(\gamma(0)\) is a limit point of the set
\(\gamma(\mathbb{Z})\). But \(\mathbb{Z}\) has no limit point in \(\mathbb{R}\), so the
corestriction \(\gamma : \mathbb{R} \to \gamma(\mathbb{R})\) cannot be a homeomorphism. In fact
the image is dense in \(\mathbb{T}^2\), an injective immersed line that winds around the
torus.
Lemma (Dirichlet's Approximation Theorem)
Given any real number \(\alpha\) and any positive integer \(N\), there exist integers \(n, m\)
with \(1 \leq n \leq N\) such that \(|n\alpha - m| \lt 1/N\).
Proof Sketch.
Let \(\{x\}\) denote the fractional part of \(x\). The \(N + 1\) numbers
\(\{0\}, \{\alpha\}, \{2\alpha\}, \dots, \{N\alpha\}\) all lie in the interval \([0, 1)\),
which splits into the \(N\) subintervals \([0, 1/N), [1/N, 2/N), \dots, [(N-1)/N, 1)\). By the
pigeonhole principle, two of them, say \(\{i\alpha\}\) and \(\{j\alpha\}\) with \(i \lt j\),
fall in the same subinterval, so \(|\{j\alpha\} - \{i\alpha\}| \lt 1/N\). Taking \(n = j - i\)
and \(m = \lfloor j\alpha \rfloor - \lfloor i\alpha \rfloor\) gives \(|n\alpha - m| \lt 1/N\)
with \(1 \leq n \leq N\).
Three independent failure modes
The three examples isolate three logically independent obstructions. The curve \((t^3, 0)\) is
a perfectly good topological embedding that fails only the differential test. Its image is the
smooth \(x\)-axis, and the defect lies in the parametrization, whose velocity vanishes at
\(t = 0\). The figure-eight is a genuine immersion and is injective. It fails only because its
image folds back to touch itself, so the domain topology and the subspace topology disagree at
the crossing. The dense torus line is an injective immersion whose image has no
self-intersections at all, yet still fails, because the image accumulates on itself globally
without ever crossing. An embedding must avoid all three pathologies at once.
When Injective Immersions Are Embeddings
The failures of the two injective immersions in the previous section, the figure-eight and the
dense line, involve global topological misbehavior. Each is a perfectly good immersion
locally. This suggests that an injective immersion should become an embedding as soon as some
hypothesis rules out the global accumulation. Several such hypotheses, each a familiar topological
condition, do the job, and they cover most cases that arise in practice.
Proposition: Sufficient Conditions for an Embedding
Suppose \(M\) and \(N\) are smooth manifolds with or without boundary, and \(F : M \to N\) is
an injective smooth immersion. If any of the following holds, then \(F\) is a smooth embedding:
(a) \(F\) is an open map or a closed map.
(b) \(F\) is a
proper map.
(c) \(M\) is compact.
(d) \(M\) has empty boundary and \(\dim M = \dim N\).
Proof Sketch.
In each case the goal is to upgrade the injective immersion into a topological embedding. The
smooth-immersion half is given.
For (a), suppose \(F\) is open or closed. The corestriction
\(F : M \to F(M)\) is a continuous bijection. The open (respectively closed) property does
transfer to this corestriction. The reason is not that restrictions of open or closed maps are
generally open or closed, which is false, but that we are restricting onto the image.
For a closed set \(C \subseteq M\), the set \(F(C)\) is closed in \(N\) by hypothesis, and
since \(F(C) \subseteq F(M)\) we have \(F(C) = F(C) \cap F(M)\), which is by definition closed
in the subspace \(F(M)\). The open case is identical. By the
characterization of homeomorphisms among continuous bijections
the corestriction is therefore a homeomorphism, so \(F\) is a topological embedding. Together
with the given smooth-immersion property, \(F\) is a smooth embedding.
Cases (b) and (c) reduce to (a) by showing \(F\) is closed.
If \(F\) is proper, then because \(N\) is
locally compact
and Hausdorff, \(F\) is closed by the theorem that
proper continuous maps are closed.
If \(M\) is compact, then \(F\) is closed by the
closed map lemma
directly. Either way (a) applies.
For (d), suppose \(M\) has empty boundary and \(\dim M = \dim N = n\). We
check first that \(F(M)\) misses \(\partial N\). Assume for contradiction that
\(F(p) \in \partial N\) for some \(p \in M\). Take a smooth chart \((U, \varphi)\) for \(M\)
around \(p\), whose image is open in \(\mathbb{R}^n\) because \(M\) is boundaryless, and a
boundary chart
\((V, \psi)\) for \(N\) around \(F(p)\), shrinking \(U\) so that \(F(U) \subseteq V\). Then
\(\psi(V)\) is open in the
upper half-space
\(\mathbb{H}^n\), and \(\psi(F(p))\) lies in \(\partial\mathbb{H}^n\). The coordinate
representation \(\widehat F = \psi \circ F \circ \varphi^{-1}\) is a smooth map from an open
subset of \(\mathbb{R}^n\) into \(\mathbb{R}^n\), and since \(F\) is an immersion its Jacobian
at \(\varphi(p)\) is an \(n \times n\) matrix of
rank
\(n\), hence invertible. By the
inverse function theorem
the image of \(\widehat F\) contains a neighborhood of \(\psi(F(p))\) in \(\mathbb{R}^n\).
That image lies in \(\psi(V) \subseteq \mathbb{H}^n\), and no subset of \(\mathbb{H}^n\)
contains a neighborhood of a point of \(\partial\mathbb{H}^n\). Hence \(F\) maps \(M\) into
the boundaryless manifold \(\operatorname{Int} N\). Between manifolds without boundary, an
immersion with equal dimensions is a
local diffeomorphism,
and such a map is open. Since \(\operatorname{Int} N\) is open in \(N\), the composite
\(M \to \operatorname{Int} N \hookrightarrow N\) is an open injective map, and (a) applies
once more.
The compact case is the one most often invoked. For instance, the inclusion
\(\iota : S^n \hookrightarrow \mathbb{R}^{n+1}\) is an injective smooth immersion. Smoothness and
injectivity of its differential are verified directly in graph coordinates. Since \(S^n\) is
compact, part (c) makes it a smooth embedding. None of these conditions is necessary, however.
There are smooth embeddings that are neither open nor closed maps.
Immersions are locally embeddings
The failures of the two injective immersions examined above were global. Locally, an immersion is
always an embedding. This is the precise sense in which the rank theorem's normal form for
immersions is a local statement about being embedded.
Theorem (Local Embedding Theorem)
Let \(M\) be a smooth manifold with or without boundary, let \(N\) be a smooth manifold
without boundary, and let \(F : M \to N\) be a smooth map. Then \(F\) is a smooth immersion if
and only if every point of \(M\) has a neighborhood \(U\) such that \(F|_U : U \to N\) is a
smooth embedding.
Proof Sketch.
One direction is immediate. If every point has a neighborhood on which \(F\) is an embedding,
then \(F\) has injective differential everywhere, so it is an immersion.
Conversely, suppose \(F\) is an immersion and fix \(p \in M\). The
rank theorem,
applied to the restriction of \(F\) to the boundaryless \(\operatorname{Int} M\) when \(p\) is
an interior point, or its
boundary counterpart
when \(p \in \partial M\), puts \(F\) into the standard immersion form near \(p\). From that
form one reads off a neighborhood \(U_1\) of \(p\) on which \(F\) is injective.
Every manifold, with or without boundary, carries a
basis of precompact coordinate balls and half-balls.
Applied to the open submanifold \(U_1\), it supplies a neighborhood \(U\) of \(p\) inside
\(U_1\) whose closure in \(U_1\) is compact. Since
compact subsets of a Hausdorff space are closed,
that compact set is closed in \(M\), so it is also the closure \(\overline U\) formed in
\(M\), and \(\overline U \subseteq U_1\). The restriction of \(F\) to the compact set
\(\overline U\) is an injective continuous map with compact domain, and therefore a
topological embedding by the
closed map lemma.
A restriction of a topological embedding is again one, so \(F|_U\) is both a topological
embedding and a smooth immersion, hence a smooth embedding.
This theorem points toward a purely topological notion. For arbitrary topological spaces \(X\) and
\(Y\), a continuous map \(F : X \to Y\) may be called a topological immersion if
every point of \(X\) has a neighborhood on which \(F\) is a topological embedding. Every smooth
immersion is a topological immersion. A topological immersion that happens to be smooth need not
be a smooth immersion, however, just as a smooth topological embedding need not be one. Recall the
curve \((t^3, 0)\).
Submersions and Local Sections
We turn from immersions to their dual, the
smooth submersions,
maps whose differential is surjective everywhere. One of the most important applications of the
rank theorem is to the theory of submersions, and the key that unlocks all of it is that
submersions admit an abundance of local right inverses. Where an immersion looks locally like an
inclusion, a submersion looks locally like a projection, and a projection can always be split by a
section.
Let \(\pi : M \to N\) be a continuous map. A section of \(\pi\) is a continuous
right inverse, a continuous map \(\sigma : N \to M\) with
\(\pi \circ \sigma = \operatorname{Id}_N\). A local section is a continuous map
\(\sigma : U \to M\) defined on some open subset \(U \subseteq N\) and satisfying
\(\pi \circ \sigma = \operatorname{Id}_U\). Sections need not exist globally, but smooth
submersions are characterized by having smooth local sections through every point.
Theorem (Local Section Theorem)
Let \(M\) and \(N\) be smooth manifolds (without boundary) and \(\pi : M \to N\) a smooth map.
Then \(\pi\) is a smooth submersion if and only if every point of \(M\) is in the image of a
smooth local section of \(\pi\).
Proof Sketch.
Suppose \(\pi\) is a smooth submersion, and let \(p \in M\), \(q = \pi(p)\). The
rank theorem
gives coordinates \((x^1, \dots, x^m)\) centered at \(p\) and \((y^1, \dots, y^n)\) centered
at \(q\) in which \(\pi\) is the projection \((x^1, \dots, x^m) \mapsto (x^1, \dots, x^n)\).
On a small coordinate cube the map
\(\sigma(x^1, \dots, x^n) = (x^1, \dots, x^n, 0, \dots, 0)\) is a smooth local section with
\(\sigma(q) = p\). This section is the coordinate splitting of the projection.
Conversely, suppose every point is in the image of a smooth local section. Given \(p \in M\),
let \(\sigma : U \to M\) be a smooth local section with \(\sigma(q) = p\), where
\(q = \pi(p)\). Differentiating \(\pi \circ \sigma = \operatorname{Id}_U\) at \(q\) and applying
the
chain rule
gives \(d\pi_p \circ d\sigma_q = \operatorname{Id}_{T_qN}\), so \(d\pi_p\) is surjective. As
this holds at every \(p\), the map \(\pi\) is a submersion.
Just as the local embedding theorem suggested a purely topological notion of immersion, this
theorem motivates a topological notion of submersion. A continuous map \(\pi : X \to Y\) is a
topological submersion if every point of \(X\) lies in the image of a continuous
local section. Every smooth submersion is a topological submersion, but not conversely.
The abundance of local sections immediately yields the first structural property of submersions.
Proposition: Submersions Are Open Quotient Maps
Let \(M\) and \(N\) be smooth manifolds (without boundary) and \(\pi : M \to N\) a smooth
submersion. Then \(\pi\) is an open map, and if it is surjective it is a
quotient map.
Proof Sketch.
Let \(W \subseteq M\) be open and let \(q \in \pi(W)\), say \(q = \pi(p)\) with \(p \in W\). By
the local section theorem there is a smooth local section \(\sigma : U \to M\) with
\(\sigma(q) = p\). The set \(\sigma^{-1}(W)\) is open in \(U\), contains \(q\), and is carried
into \(\pi(W)\) by the identity \(y = \pi(\sigma(y))\) for \(y \in \sigma^{-1}(W)\). Thus every
point of \(\pi(W)\) has an open neighborhood inside \(\pi(W)\), so \(\pi(W)\) is open and
\(\pi\) is an open map. A surjective open continuous map is a quotient map.
Smooth Quotients
In topology, a surjective quotient map lets one transfer continuity questions from a quotient
space back to the space above it. A map out of the quotient is continuous exactly when its
composition with the quotient map is. Surjective smooth submersions play precisely this role in
the smooth category. The three theorems of this section are the smooth analogues of the
corresponding facts for topological quotient maps, and together they show that a surjective smooth
submersion exhibits its codomain as a smooth quotient of its domain, uniquely up to
diffeomorphism.
Theorem (Characteristic Property of Surjective Smooth Submersions)
Suppose \(M\) and \(N\) are smooth manifolds (without boundary) and \(\pi : M \to N\) is a
surjective smooth submersion. For any smooth manifold \(P\), with or without boundary, a map
\(F : N \to P\) is smooth if and only if \(F \circ \pi : M \to P\) is smooth.
Proof Sketch.
If \(F\) is smooth, then \(F \circ \pi\) is smooth as a composition. Conversely, suppose
\(F \circ \pi\) is smooth, and let \(q \in N\). Surjectivity gives a point
\(p \in \pi^{-1}(q)\), and the
local section theorem
provides a smooth local section \(\sigma : U \to M\) with \(\sigma(q) = p\). On \(U\),
\[
F|_U = F|_U \circ \operatorname{Id}_U = F|_U \circ (\pi \circ \sigma) = (F \circ \pi) \circ \sigma,
\]
a composition of smooth maps, so \(F\) is smooth in a neighborhood of each point of \(N\),
hence smooth.
The property is called characteristic because \(\pi\) and \(M\) alone determine \(N\) up
to diffeomorphism, a point made precise by the uniqueness theorem below. Its most frequent use is
to manufacture smooth maps out of a quotient. If a smooth map on \(M\) is constant on the fibers
of \(\pi\), it descends.
Theorem (Passing Smoothly to the Quotient)
Suppose \(M\) and \(N\) are smooth manifolds (without boundary) and \(\pi : M \to N\) is a
surjective smooth submersion. If \(P\) is a smooth manifold, with or without boundary, and
\(F : M \to P\) is a smooth map that is constant on the fibers of \(\pi\), then there exists a
unique smooth map \(\widetilde F : N \to P\) such that \(\widetilde F \circ \pi = F\).
Proof Sketch.
Because \(F\) is constant on the fibers of \(\pi\) and \(\pi\) is surjective, there is a unique
function \(\widetilde F : N \to P\) with \(\widetilde F \circ \pi = F\). Since
\(\widetilde F \circ \pi = F\) is smooth, the characteristic property above makes
\(\widetilde F\) smooth.
Finally, the quotient is unique. Any two surjective smooth submersions out of \(M\) with the same
fibers present the same smooth manifold.
Theorem (Uniqueness of Smooth Quotients)
Suppose \(M\), \(N_1\), and \(N_2\) are smooth manifolds (without boundary), and
\(\pi_1 : M \to N_1\), \(\pi_2 : M \to N_2\) are surjective smooth submersions that are
constant on each other's fibers. Then there exists a unique diffeomorphism \(F : N_1 \to N_2\)
such that \(F \circ \pi_1 = \pi_2\).
Proof Sketch.
The hypothesis that each map is constant on the other's fibers says exactly that \(\pi_1\) and
\(\pi_2\) induce the same partition of \(M\):
\(\pi_1(x) = \pi_1(y) \iff \pi_2(x) = \pi_2(y)\).
Because \(\pi_2\) is constant on the fibers of \(\pi_1\), passing to the quotient produces a
unique smooth map \(F : N_1 \to N_2\) with \(F \circ \pi_1 = \pi_2\). Symmetrically, \(\pi_1\)
constant on the fibers of \(\pi_2\) produces a smooth \(G : N_2 \to N_1\) with
\(G \circ \pi_2 = \pi_1\). Then \(G \circ F \circ \pi_1 = \pi_1\) and
\(F \circ G \circ \pi_2 = \pi_2\). Since \(\pi_1, \pi_2\) are surjective, \(G \circ F\) and
\(F \circ G\) are the respective identities, so \(F\) is a diffeomorphism with inverse \(G\).
Submersions as the smooth face of quotient maps
These three theorems mirror, one for one, the basic facts about topological quotient maps. The
characteristic property is the smooth counterpart of the statement that a map out of a
quotient is continuous exactly when its composite with the quotient map is continuous. Passing
to the quotient is the smooth descent of fiber-constant maps. Uniqueness says the quotient is
determined by its fibers alone.
The upshot is a working principle for the rest of the theory. Whenever a smooth manifold is
built as a set of equivalence classes, exhibiting the projection as a surjective smooth
submersion pins the smooth structure down uniquely and guarantees that maps respecting the
identification descend to it. This is the mechanism behind the smooth structures on projective
spaces, Grassmannians, and homogeneous spaces, and it is the submersion side of the duality
that the pages ahead develop alongside the embedding side.