Smooth Embeddings
A
smooth immersion
is a map whose differential is injective at every point — a purely local, purely differential
condition. 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 — same points, same topology, same
smooth structure — we must add a topological condition to the differential one.
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
— that is, 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 immersion 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 \(\gamma : \mathbb{R} \to \mathbb{R}^2\),
\(\gamma(t) = (t^3, 0)\), is smooth and is a topological embedding — it is a homeomorphism onto the
\(x\)-axis. Yet it is not a smooth embedding, because \(\gamma'(0) = 0\), 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 of injectivity of the topology — the figure-eight. 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 — a dense curve on the torus. 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; and it is injective, 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 densely 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 \(|n\alpha - m| < 1/N\).
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 \(i < j\), fall
in the same subinterval, so \(|\{j\alpha\} - \{i\alpha\}| < 1/N\). Taking \(n = j - i\) and
\(m = \lfloor j\alpha \rfloor - \lfloor i\alpha \rfloor\) gives \(|n\alpha - m| < 1/N\) with
\(1 \leq n \leq N\). \(\blacksquare\)
Three independent failure modes
The three examples isolate three logically independent obstructions. The cusped curve
\((t^3, 0)\) is a perfectly good topological embedding that fails only the differential test —
the image has a hidden singularity invisible to the topology. The figure-eight is a genuine
immersion and injective, failing 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 is precisely a map
that avoids all three pathologies at once.
When Injective Immersions Are Embeddings
The three pathologies of the previous section all involve global topological misbehavior;
each example 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 — not because restrictions of open or closed maps are
generally open or closed, which is false, but because 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), the equidimensional hypothesis makes \(F\) a local diffeomorphism: with
empty boundary and equal dimensions, an immersion is a
local diffeomorphism,
and a local diffeomorphism is an open map. Being open, \(F\) carries the boundaryless \(M\) into
the interior of \(N\) — a point of \(\partial N\) has no neighborhood that is open in \(N\), so it
cannot lie in the open set \(F(M)\). Thus the composite
\(M \to \operatorname{Int} N \hookrightarrow N\) is an open injective map, and (a)
applies once more. \(\blacksquare\)
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 — and 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 examples of the previous section show that immersions fail to be embeddings only for global
reasons. 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 \(M\) and \(N\) be smooth manifolds with or 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
(at an interior point) or its
boundary counterpart
(at a boundary point) puts \(F\) into the standard immersion form near \(p\), from which one
reads off a neighborhood \(U_1\) on which \(F\) is injective. Now choose a
precompact neighborhood
\(U\) of \(p\) with \(\overline U \subseteq U_1\). The restriction of \(F\) to the compact set
\(\overline U\) is an injective continuous map with compact domain, hence a topological
embedding by the
closed map lemma;
and a restriction of a topological embedding is again one,
so \(F|_U\) is a topological embedding and a smooth immersion — a smooth embedding.
\(\blacksquare\)
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; but, just as a smooth topological embedding need not be a
smooth immersion — recall the cusped curve \((t^3, 0)\) — a topological immersion that happens to
be smooth need not be a smooth immersion.
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 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 \(\sigma(q) = p\): it 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. \(\blacksquare\)
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 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. \(\blacksquare\)
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 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. \(\blacksquare\)
The property is called characteristic because it determines \(N\), up to diffeomorphism,
from \(\pi\) and \(M\) alone — 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 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.
A surjective smooth submersion is a
quotient map,
so — regarding its fibers as the classes of an equivalence relation on \(M\) — the
universal property of the quotient
produces a unique continuous map \(\widetilde F : N \to P\) with
\(\widetilde F \circ \pi = F\) — the hypothesis that \(F\) is constant on fibers is exactly what
makes \(\widetilde F\) well defined as a function. Since \(\widetilde F \circ \pi = F\) is
smooth, the characteristic property above upgrades \(\widetilde F\) from continuous to smooth.
\(\blacksquare\)
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, 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 \(G\).
\(\blacksquare\)
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 it lifts continuously; passing to the quotient is the smooth descent
of fiber-constant maps; and 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 — orbits of a group action, lines through the origin, points
identified by a symmetry — exhibiting the projection as a surjective smooth submersion is
what certifies that the construction carries a well-defined smooth structure, and 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 next stage of the theory develops alongside the embedding side.