Embedded Submanifolds
We have already met one way for a manifold to sit inside another. When \(U \subseteq M\) is an
open submanifold,
it inherits a smooth structure from \(M\) and occupies a full-dimensional open piece of it. That
construction, we noted at the time, was only the easiest case of a far more general notion. That
notion allows a submanifold to have lower dimension than the ambient space, like a curve in
the plane or a sphere in space. We now develop that general notion.
The right definition is not a set-theoretic one. A subset of \(M\) carries no smooth structure on
its own. What we ask instead is that the subset, given a smooth structure of its own, sit inside
\(M\) as faithfully as possible. Concretely, its inclusion must be a
smooth embedding.
Definition: Embedded Submanifold
Let \(M\) be a smooth manifold. An embedded submanifold of \(M\) is a subset
\(S \subseteq M\) that is a manifold (without boundary) in the subspace topology, endowed with a
smooth structure for which the inclusion map \(\iota : S \hookrightarrow M\) is a smooth
embedding. The codimension of \(S\) in \(M\) is the difference
\(\dim M - \dim S\), and \(M\) is called the ambient manifold. An embedded
submanifold of codimension \(1\) is called an embedded hypersurface.
Two features of this definition deserve immediate emphasis. First, the topology on \(S\) is not
chosen freely. The subspace topology inherited from \(M\) is the only admissible one. This is
exactly what distinguishes an embedded submanifold from the more permissive notion we take up at the
end of this page, where the topology may be finer than the subspace topology. Second, an embedded
submanifold is by definition a manifold without boundary. Through the first four sections
the ambient manifold \(M\) is likewise taken without boundary, so that every map in sight is of
the interior kind. The theorem on boundaries below is the one place before the final section where
\(M\) is allowed a boundary, and the definitions of this section are read there verbatim. The
final section lifts the restriction on both sides at once, allowing the submanifold and the ambient
manifold alike to carry a boundary. The empty set qualifies, vacuously, as an
embedded submanifold of any dimension.
The definition is abstract, but its content is captured by a single, very concrete source of
examples: the image of a smooth embedding. Indeed, the inclusion of an embedded submanifold is
itself an embedding, so every embedded submanifold arises this way. Conversely, every embedding
produces one.
Proposition: Images of Embeddings Are Submanifolds
Let \(N\) and \(M\) be smooth manifolds and let \(F : N \to M\) be a smooth embedding. Then
\(S = F(N)\), with the subspace topology, is a topological manifold, and it has a unique smooth
structure making it an embedded submanifold of \(M\) for which \(F\) is a diffeomorphism onto
\(S\).
Proof Sketch.
Because \(F\) is a topological embedding, its corestriction \(F : N \to S\) is a homeomorphism,
so \(S\) is a topological manifold of the same dimension as \(N\). Local Euclidean structure
transports across the homeomorphism, while Hausdorffness and second-countability pass to \(S\)
as a subspace of \(M\). We transport the smooth structure across this homeomorphism by declaring
a chart of \(S\) to be \((F(U), \varphi \circ F^{-1})\) for each chart \((U, \varphi)\) of
\(N\). The transition maps of these charts coincide with those of \(N\), so they form a smooth
atlas, and with it \(F : N \to S\) is a diffeomorphism by construction. The inclusion then
factors as
\[
\iota : S \xrightarrow{\ F^{-1}\ } N \xrightarrow{\ F\ } M,
\]
a diffeomorphism followed by the embedding \(F\), hence is itself a smooth embedding. This
smooth structure therefore makes \(S\) an embedded submanifold. Uniqueness holds because any
smooth structure on \(S\) for which \(F\) is a diffeomorphism must have exactly these charts.
This proposition is the working characterization of embedded submanifolds, which are precisely the
images of smooth embeddings. The remainder of this section assembles a catalogue of such images,
each obtained by exhibiting an explicit embedding. We begin with the constructions that recur most
often in practice.
The simplest are the open submanifolds themselves. If \(S \subseteq M\) is open, the inclusion is a
smooth embedding of full rank, so \(S\) is an embedded submanifold of
codimension \(0\). The converse also holds. An embedded submanifold of codimension
\(0\) is an open subset. Its inclusion is an embedding, hence an immersion, and between manifolds of
equal dimension an immersion is a
local diffeomorphism,
and a local diffeomorphism is an
open map,
so its image is open. Thus the embedded submanifolds of codimension \(0\) are exactly the open
submanifolds, the case we singled out earlier. The genuinely new content of this page lies in
positive codimension.
Proposition: Slices of Product Manifolds
Let \(M\) and \(N\) be smooth manifolds and let \(p \in N\). The subset \(M \times \{p\}\),
called a slice of the product, is an embedded submanifold of \(M \times N\),
diffeomorphic to \(M\).
Proof Sketch.
The map \(x \mapsto (x, p)\) is a smooth embedding of \(M\) into \(M \times N\). It is a smooth
immersion, being a section of the projection \(\pi_M\), and a homeomorphism onto its image,
whose inverse is the restriction of \(\pi_M\). Its image is \(M \times \{p\}\), so the previous
proposition applies.
The most flexible construction realizes a submanifold as the graph of a smooth map. Graphs
will reappear throughout this page. They are the local model for every embedded submanifold, and the
link between submanifolds and the equations that cut them out.
Proposition: Graphs Are Embedded Submanifolds
Let \(M\) be a smooth \(m\)-manifold and \(N\) a smooth \(n\)-manifold, let \(U \subseteq M\) be
open, and let \(f : U \to N\) be smooth. Then the graph
\[
\Gamma(f) = \{(x, y) \in M \times N : x \in U,\ y = f(x)\}
\]
is an embedded \(m\)-dimensional submanifold of \(M \times N\), diffeomorphic to \(U\).
Proof Sketch.
Consider the map \(\gamma_f : U \to M \times N\), \(\gamma_f(x) = (x, f(x))\). It is smooth, and
the projection \(\pi_M : M \times N \to M\) satisfies
\(\pi_M \circ \gamma_f = \operatorname{Id}_U\). Differentiating this identity and applying the
chain rule
gives \(d(\pi_M)_{\gamma_f(x)} \circ d(\gamma_f)_x = \operatorname{Id}_{T_xM}\), so each
\(d(\gamma_f)_x\) has a left inverse and is therefore injective. Thus \(\gamma_f\) is a smooth
immersion. The same identity \(\pi_M \circ \gamma_f = \operatorname{Id}_U\) shows that the
continuous map \(\pi_M\), restricted to \(\Gamma(f)\), inverts the corestriction
\(\gamma_f : U \to \Gamma(f)\). Hence that corestriction is a homeomorphism and \(\gamma_f\) is
a topological embedding. Being both, \(\gamma_f\) is a smooth embedding with image
\(\Gamma(f)\), which is therefore an embedded \(m\)-dimensional submanifold diffeomorphic to
\(U\).
Properly Embedded Submanifolds
The examples so far are embedded but may still be badly placed in the ambient manifold. An open
interval, embedded as an open arc, is an embedded submanifold of the plane, yet its closure adds
endpoints that do not belong to it. Many later constructions require a submanifold to be closed as a
subset, and this turns out to be equivalent to a clean condition on the inclusion map. Restricting
and extending smooth functions are among them.
Definition: Properly Embedded Submanifold
An embedded submanifold \(S \subseteq M\) is properly embedded if the inclusion
\(\iota : S \hookrightarrow M\) is a
proper map.
That is, the preimage of every compact set is compact.
Proposition: Proper Embedding and Closedness
An embedded submanifold \(S \subseteq M\) is properly embedded if and only if it is a closed
subset of \(M\). In particular, every compact embedded submanifold is properly embedded.
The graph construction above produces only an embedded submanifold, because its domain is an open
subset \(U\). The embedding can fail to be proper near the boundary of \(U\). When the domain is the
entire manifold, however, the graph is automatically properly embedded. The distinction is
worth isolating, since it is the global version that arises whenever a submanifold is presented as
the graph of a globally defined map.
Proposition: Global Graphs Are Properly Embedded
Let \(M\) and \(N\) be smooth manifolds and let \(f : M \to N\) be smooth, defined on all of
\(M\). With the smooth structure of the previous proposition, the graph \(\Gamma(f)\) is a
properly embedded submanifold of \(M \times N\).
Proof Sketch.
The embedding \(\gamma_f : M \to M \times N\) has the projection \(\pi_M\) as a continuous left
inverse, since \(\pi_M \circ \gamma_f = \operatorname{Id}_M\). A continuous map into a Hausdorff
space with a continuous left inverse is
proper,
so \(\gamma_f\) is proper. Its image \(\Gamma(f)\) is therefore a properly embedded submanifold.
One source of properly embedded submanifolds deserves to be singled out, because it will underlie
the entire theory of integration over a region: the boundary of a
smooth manifold with boundary.
The boundary points were defined chart by chart, and the resulting set carries the subspace
topology. The next theorem records that this set is not merely a topological remnant but a smooth
submanifold in its own right, sitting in the ambient manifold in the cleanest possible way.
Theorem: The Boundary Is a Properly Embedded Hypersurface
Let \(M\) be a smooth \(n\)-manifold with boundary. With the subspace topology, the
boundary
\(\partial M\) is a topological \((n-1)\)-manifold without boundary, and it admits a smooth
structure with respect to which it is a properly embedded submanifold of \(M\) of codimension
one. A submanifold of codimension one is called a hypersurface.
Proof Sketch.
Each
boundary chart
\((U, \varphi)\) for \(M\) carries \(U \cap \partial M\) to
\(\partial \mathbb{H}^n = \{x \in \mathbb{H}^n : x^n = 0\}\), an ordinary \((n-1)\)-dimensional
slice of the model half-space. Discarding the last coordinate, the restriction
\(\varphi|_{U \cap \partial M}\) followed by the identification
\(\partial \mathbb{H}^n \approx \mathbb{R}^{n-1}\) gives a chart for \(\partial M\). Two such
charts overlap smoothly because they are restrictions of the smoothly compatible boundary charts
of \(M\). These charts cover \(\partial M\) and turn it into a smooth \((n-1)\)-manifold without
boundary, since the slice coordinates range over an open subset of \(\mathbb{R}^{n-1}\) with no
half-space constraint.
With respect to these charts the inclusion \(\partial M \hookrightarrow M\) has, in every
boundary chart, the coordinate representation \(x' \mapsto (x', 0)\), the inclusion of a
coordinate slice. This is the local model of a smooth embedding, so the inclusion is a smooth
embedding and \(\partial M\) is an embedded submanifold of codimension one. (The local slice
criterion developed in the next section is stated for submanifolds of a manifold without
boundary. Here the same slice picture is supplied directly by the boundary charts of \(M\), so
no separate criterion for the boundary case is needed.) Finally, \(\partial M\) is closed in
\(M\). Its complement is the
interior
\(\operatorname{Int} M\), which is open because every interior point lies in the domain of an
interior chart contained in \(\operatorname{Int} M\). A closed embedded submanifold is
properly embedded,
which completes the proof.
Slice Charts and the Local Slice Criterion
The catalogue of the previous section produces embedded submanifolds by exhibiting embeddings, but
it leaves a basic recognition problem unsolved: given a subset \(S \subseteq M\), how can we tell
whether it is an embedded submanifold, without first guessing the manifold structure it
ought to carry? The answer is a local criterion, phrased entirely in terms of the ambient charts of
\(M\), that makes no reference to any topology or smooth structure on \(S\) in advance. It rests on
the model picture an immersion always realizes locally, the inclusion of a coordinate subspace.
Identify \(\mathbb{R}^k\) with the subset of \(\mathbb{R}^n\) where the last \(n - k\) coordinates
vanish. More generally, a \(k\)-slice of an open set \(U \subseteq \mathbb{R}^n\)
is a subset of the form
\[
\{(x^1, \dots, x^n) \in U : x^{k+1} = c^{k+1}, \dots, x^n = c^n\}
\]
for fixed constants \(c^{k+1}, \dots, c^n\). Geometrically, freezing the last \(n - k\) coordinates
at constant values and letting the first \(k\) range freely cuts out a flat \(k\)-dimensional sheet
sitting inside the \(n\)-dimensional box. A single horizontal plane \(\{z = c\}\) sits in
\(\mathbb{R}^3\) the same way. Accordingly, each such slice is homeomorphic to an open subset of
\(\mathbb{R}^k\) under the first \(k\) coordinates. The definition transfers verbatim to a manifold
through a chart.
Definition: Slice Chart and the Local Slice Condition
Let \(M\) be a smooth \(n\)-manifold and \(S \subseteq M\) a subset. A smooth chart
\((U, \varphi)\) of \(M\) is a slice chart for \(S\) (of dimension \(k\)) if
\(\varphi(S \cap U)\) is a \(k\)-slice of \(\varphi(U)\). The coordinates \((x^1, \dots, x^n)\)
of such a chart are called slice coordinates. The subset \(S\) is said to
satisfy the local \(k\)-slice condition if every point of \(S\) is contained in
the domain of a slice chart for \(S\) of dimension \(k\). This is a condition on the subset
\(S\) alone, presupposing no topology or smooth structure on it.
By subtracting the constants \(c^{k+1}, \dots, c^n\) from the corresponding coordinate functions, we
may always arrange that a slice chart presents \(S \cap U\) as the slice through the origin, where
\(x^{k+1} = \dots = x^n = 0\). We do so freely below. The decisive fact is that this purely
set-theoretic condition is equivalent to being an embedded submanifold, and that when it holds, the
submanifold structure is forced.
Theorem (Local Slice Criterion for Embedded Submanifolds)
Let \(M\) be a smooth \(n\)-manifold. A subset \(S \subseteq M\) is an embedded
\(k\)-dimensional submanifold if and only if \(S\) satisfies the local \(k\)-slice condition.
Moreover, when \(S\) satisfies the local \(k\)-slice condition, the smooth structure making it
an embedded submanifold is uniquely determined. It is the one for which the slice charts of
\(M\) restrict to charts of \(S\).
Proof Sketch.
Suppose first that \(S\) is an embedded \(k\)-submanifold, and let \(p \in S\). The inclusion
\(\iota : S \hookrightarrow M\) is a smooth immersion, hence of constant rank \(k\). The
rank theorem
therefore supplies a chart \((V_0, \varphi)\) of \(M\) centered at \(p\) and a chart of \(S\)
centered at \(p\) in which \(\iota\) has the coordinate representation
\((x^1, \dots, x^k) \mapsto (x^1, \dots, x^k, 0, \dots, 0)\). Thus a neighborhood of \(p\) in
\(S\) is carried by \(\varphi\) onto an open subset of the \(k\)-slice
\(\{x^{k+1} = \dots = x^n = 0\}\). Because \(S\) carries the subspace topology, that
neighborhood is \(W \cap S\) for some open \(W \subseteq M\). Intersecting \(V_0\) with \(W\)
yields a slice chart for \(S\) about \(p\). Hence \(S\) satisfies the local \(k\)-slice
condition.
Conversely, suppose \(S\) satisfies the local \(k\)-slice condition, and give \(S\) the subspace
topology. As a subspace of a manifold, \(S\) is
Hausdorff and second-countable.
It remains to produce charts. Given a slice chart \((U, \varphi)\) with \(\varphi(S \cap U)\)
the slice \(\{x^{k+1} = \dots = x^n = 0\}\), let \(\pi : \mathbb{R}^n \to \mathbb{R}^k\) be the
projection onto the first \(k\) coordinates and set \(\psi = \pi \circ \varphi|_{S \cap U}\). We
claim \(\psi\) maps \(S \cap U\) homeomorphically onto an open subset of \(\mathbb{R}^k\). Its
image \(\psi(S \cap U) = \pi(\varphi(S \cap U))\) is the projection of the slice
\(\varphi(S \cap U)\), hence open in \(\mathbb{R}^k\), and its inverse is
\(\varphi^{-1} \circ j\), where \(j(x^1, \dots, x^k) = (x^1, \dots, x^k, 0, \dots, 0)\). Since
both \(\psi\) and this inverse are continuous, \(\psi\) is a homeomorphism. Two such charts have
transition map \(\psi' \circ \psi^{-1} = \pi \circ \varphi' \circ \varphi^{-1} \circ j\), a
composition of smooth maps between open subsets of Euclidean spaces, so the charts are smoothly
compatible and form a smooth atlas.
In these coordinates the inclusion is the standard slice inclusion
\((x^1, \dots, x^k) \mapsto (x^1, \dots, x^k, 0, \dots, 0)\), which is a smooth immersion, and
it is a
topological embedding
because \(S\) carries the subspace topology. The inclusion is therefore a smooth embedding and
\(S\) is an embedded \(k\)-submanifold. The uniqueness asserted in the statement is not
established by this argument. What is proved later is in fact stronger. No other
topology or smooth structure on \(S\) renders the inclusion an embedding or even an immersion.
That statement rests on the global rank theorem and is settled once we have the tools to
restrict maps to submanifolds, where the
uniqueness
is proved in full.
The criterion delivers at once the example that has accompanied the manifold series from its start.
The sphere \(\mathbb{S}^n \subseteq \mathbb{R}^{n+1}\) meets each open half-space
\(\{x : x^i \gt 0\}\) in the graph of the smooth function
\(x^i = \sqrt{1 - \sum_{j \neq i} (x^j)^2}\), and each \(\{x : x^i \lt 0\}\) in the graph of its
negative. These graphs are exactly the \(n\)-slices of suitable charts, and together they cover
\(\mathbb{S}^n\). So \(\mathbb{S}^n\) satisfies the local \(n\)-slice condition and is an embedded
hypersurface.
By the uniqueness clause, the structure the criterion produces is the only one making
\(\mathbb{S}^n\) a submanifold. It is precisely the
standard smooth structure
we once assembled by hand from graph coordinates, whose charts are the very slice charts just
described. What was a construction is now a consequence.
Level Sets
In practice, embedded submanifolds are most often presented not by parametrizations but by
equations: the unit sphere is \(\{|x|^2 = 1\}\), a level curve is \(\{f(x,y) = c\}\), and
the configuration space of a mechanism is cut out by its constraints. Given any map
\(\Phi : M \to N\) and a point \(c \in N\), the level set \(\Phi^{-1}(c)\) is the
solution set of the equation \(\Phi = c\). When \(N = \mathbb{R}^k\) and \(c = 0\) it is the
zero set. The question this section answers is: when is a level set an embedded
submanifold?
Not always. The three functions \(\mathbb{R}^2 \to \mathbb{R}\),
\[
\Theta(x, y) = x^2 - y, \quad \Phi(x, y) = x^2 - y^2, \quad \Psi(x, y) = x^2 - y^3,
\]
have zero sets of utterly different character. The zero set of \(\Theta\) is a parabola, the graph
of \(x \mapsto x^2\), hence a submanifold. The zero set of \(\Phi\) is the pair of crossing lines
\(y = \pm x\), which is not a submanifold at the origin, and that of \(\Psi\) is a cusped curve,
also singular there.
A level set is only as good as the map that defines it. The decisive hypothesis turns out to be a
condition on the differential. The rank theorem supplies it.
Theorem (Constant-Rank Level Set Theorem)
Let \(M\) and \(N\) be smooth manifolds and let \(\Phi : M \to N\) be a smooth map of constant
rank \(r\). Then each level set of \(\Phi\) is a properly embedded submanifold of codimension
\(r\) in \(M\).
Proof Sketch.
Write \(m = \dim M\) and \(k = m - r\). Fix \(c \in N\) and set
\(S = \Phi^{-1}(c)\). For each \(p \in S\), the
rank theorem
provides charts \((U, \varphi)\) centered at \(p\) and \((V, \psi)\) centered at \(c\) in which
\(\Phi\) has the coordinate representation
\((x^1, \dots, x^m) \mapsto (x^1, \dots, x^r, 0, \dots, 0)\). Because the charts are centered at
\(p\) and \(c\), the value \(c\) sits at the origin of the target coordinates, so a point of
\(U\) maps to \(c\) exactly when its first \(r\) coordinates vanish. In these coordinates the
level set through \(p\) is therefore exactly the set \(\{x^1 = \dots = x^r = 0\}\). Composing
\(\varphi\) with the permutation that moves \(x^1, \dots, x^r\) into the last \(r\) slots turns
this set into a \(k\)-slice, so \(p\) lies in the domain of a slice chart for \(S\). As \(p\)
was arbitrary, \(S\) satisfies the local \(k\)-slice condition and is, by the
local slice criterion,
an embedded \(k\)-submanifold of codimension \(r\). Finally, \(S = \Phi^{-1}(c)\) is closed by
continuity, hence
properly embedded.
The constant-rank hypothesis is automatic in the most important special case, that of a submersion,
whose rank is constant and equal to the dimension of the codomain.
Corollary (Submersion Level Set Theorem)
If \(\Phi : M \to N\) is a smooth submersion, then each level set of \(\Phi\) is a properly
embedded submanifold of codimension equal to \(\dim N\).
Proof Sketch.
A
smooth submersion
has constant rank equal to \(\dim N\), so the previous theorem applies with \(r = \dim N\).
The nonlinear rank-nullity law
The submersion level set theorem is the nonlinear shadow of a fact from linear algebra. A
surjective linear map \(L : \mathbb{R}^m \to \mathbb{R}^r\) has, by the
rank-nullity law,
a kernel of codimension \(r\). The equation \(Lx = 0\) imposes \(r\) independent scalar
conditions, each cutting one degree of freedom from \(\mathbb{R}^m\). A smooth submersion is the
manifold analogue of a surjective linear map, since its differential is surjective at every
point. Each of its \(r\) local component functions likewise removes one dimension, and so leaves
a level set of codimension \(r\). The kernel of a linear surjection becomes the level set of a
smooth submersion. The linear subspace becomes a submanifold.
The corollary can be sharpened. To conclude that a particular level set is a submanifold, we need
the submersion condition only on that level set, not on all of \(M\). This is the content
of the most-used version of the theorem, and it requires a vocabulary for points and values at
which the differential is surjective.
Definition: Regular and Critical Points and Values
Let \(\Phi : M \to N\) be a smooth map. A point \(p \in M\) is a regular point
of \(\Phi\) if \(d\Phi_p : T_pM \to T_{\Phi(p)}N\) is surjective, and a
critical point otherwise. A point \(c \in N\) is a
regular value of \(\Phi\) if every point of the level set \(\Phi^{-1}(c)\) is a
regular point. If instead \(\Phi^{-1}(c)\) contains at least one critical point, then \(c\) is a
critical value. A value \(c\) whose level set \(\Phi^{-1}(c)\) is empty
therefore counts as a regular value, the condition holding vacuously. A
regular level set is a level set consisting entirely of regular points, that
is, the level set of a regular value.
The whole map \(\Phi\) is a submersion precisely when every point is regular. A regular value
relaxes this to hold only along one fiber. Since
surjectivity of the differential is an open condition,
the regular points form an open set, and this is exactly what lets the local argument go through.
Corollary (Regular Level Set Theorem)
Every regular level set of a smooth map between smooth manifolds is a properly embedded
submanifold whose codimension is equal to the dimension of the codomain.
Proof Sketch.
Let \(c\) be a regular value of \(\Phi : M \to N\), and let
\(U = \{p \in M : d\Phi_p \text{ is surjective}\}\). By the openness of full rank, \(U\) is
open, and by hypothesis \(\Phi^{-1}(c) \subseteq U\). The restriction \(\Phi|_U : U \to N\) is a
submersion, so the submersion level set theorem makes \(\Phi^{-1}(c)\) an embedded submanifold
of \(U\), of codimension \(\dim N\). Being embedded in the open submanifold \(U\) and closed in
\(M\) by continuity, it is a properly embedded submanifold of \(M\). (If \(\Phi^{-1}(c)\) is
empty, it is vacuously a properly embedded submanifold, of unconstrained dimension.)
With this in hand the sphere reappears, now by its simplest proof of all. Let
\(f : \mathbb{R}^{n+1} \to \mathbb{R}\) be \(f(x) = |x|^2\). Then \(df_x(v) = 2\sum_i x^i v^i\),
which is surjective for every \(x \neq 0\). Hence \(1\) is a regular value of \(f\), and
\(\mathbb{S}^n = f^{-1}(1)\) is a properly embedded hypersurface. The same sphere we built by graph
charts, then recognized through the slice criterion, now falls out of a one-line computation. Each
pass through the theory trades construction for consequence.
Not every embedded submanifold is globally a level set of a submersion, but the next proposition
shows that every one is locally of this form, and supplies the language for the converse
direction.
Proposition: Embedded Submanifolds Are Locally Level Sets
Let \(S\) be a subset of a smooth \(m\)-manifold \(M\). Then \(S\) is an embedded
\(k\)-submanifold if and only if every point of \(S\) has a neighborhood \(U\) in \(M\) such
that \(U \cap S\) is a level set of a smooth submersion \(\Phi : U \to \mathbb{R}^{m-k}\). Such
a \(\Phi\) is called a local defining map for \(S\). When a single submersion
\(\Phi : M \to \mathbb{R}^{m-k}\) has \(S\) as a regular level set, it is a
(global) defining map. Since the codomain here is a Euclidean space, \(\Phi\)
is also called a defining function.
Proof Sketch.
If \(S\) is an embedded \(k\)-submanifold, a slice chart \((U, \varphi)\) about a point of \(S\)
presents \(U \cap S\) as \(\{x^{k+1} = \dots = x^m = 0\}\). The map
\(\Phi = (x^{k+1}, \dots, x^m) : U \to \mathbb{R}^{m-k}\), being the last \(m - k\) coordinate
functions, is a submersion with \(U \cap S = \Phi^{-1}(0)\). Conversely, if each \(U \cap S\) is
a level set of a submersion \(\Phi : U \to \mathbb{R}^{m-k}\), the
submersion level set theorem
makes each \(U \cap S\) an embedded submanifold of \(U\). Hence \(S\) satisfies the local slice
condition and is an embedded submanifold of \(M\).
Finding a defining function in a concrete case is a matter of encoding the geometry as an equation.
A surface of revolution illustrates the pattern. Let \(C\) be an embedded curve in the half-plane
\(\{(r, z) : r \gt 0\}\) cut out locally by \(g(r, z) = 0\), and revolve it about the
\(z\)-axis.
The resulting surface \(S_C = \{(x, y, z) : g(\sqrt{x^2 + y^2},\, z) = 0\}\) is then the level
set of \(\Phi(x, y, z) = g(\sqrt{x^2 + y^2},\, z)\), a smooth map on the complement of the
\(z\)-axis. There the map \((x, y, z) \mapsto (\sqrt{x^2 + y^2},\, z)\) is a submersion onto the
half-plane, so by the chain rule \(d\Phi\) is surjective wherever \(dg\) is. Thus where \(g\)
defines \(C\) regularly, \(\Phi\) defines \(S_C\) regularly, and so exhibits the surface
as an embedded submanifold of \(\mathbb{R}^3\). The doughnut-shaped torus, obtained by revolving the
circle \((r - 2)^2 + z^2 = 1\), is the regular level set of
\(\Phi(x, y, z) = (\sqrt{x^2 + y^2} - 2)^2 + z^2\) at the value \(1\).
Why data is expected to lie on a submanifold
The level set theorems give precise meaning to a working assumption that pervades modern data
analysis. High-dimensional data rarely fill their ambient space \(\mathbb{R}^n\). Images, sensor
readings, and the activations of a network cluster instead near a much lower-dimensional set,
because the data is generated by comparatively few underlying degrees of freedom subject to many
constraints. Each independent constraint is, locally, the vanishing of a smooth function, and a
family of \(r\) such constraints with surjective combined differential cuts out, by the regular
level set theorem, a submanifold of codimension \(r\).
Two intuitions run together here: that data occupies a low-dimensional surface inside a
high-dimensional space, and that learning a representation means recovering that surface. In
this language both amount to the statement that the data lies on or near an embedded submanifold
of \(\mathbb{R}^n\). The
manifold viewpoint on data
is, to that extent, made precise by the constructions of this page. Whether real data in fact
lies on such a surface is an empirical question. The language in which the assumption is even
stated, a low-dimensional submanifold cut out by constraints, is exactly the one developed here.
One task remains. Such a submanifold can always be situated inside a Euclidean space of
controlled dimension, and the manifold series takes up that embedding theory later.
Immersed Submanifolds
Every submanifold so far has carried the subspace topology. But the manifold series will repeatedly
meet subsets that behave like submanifolds locally yet are wound through the ambient space in a way
the subspace topology cannot capture. Lie subgroups, the leaves of foliations, and the image of a
curve that returns arbitrarily close to itself are all of this kind. To accommodate them we relax
the definition, keeping the differential condition while surrendering the topological one.
Definition: Immersed Submanifold
Let \(M\) be a smooth manifold. An immersed submanifold of \(M\) is a subset
\(S \subseteq M\) endowed with a topology, not necessarily the subspace topology, with
respect to which it is a topological manifold, together with a smooth structure with respect to
which the inclusion \(\iota : S \hookrightarrow M\) is a smooth immersion. Its
codimension is \(\dim M - \dim S\). Every embedded submanifold is an immersed
submanifold. The embedded ones are exactly those for which the topology happens to be the
subspace topology and the inclusion an embedding.
A word on terminology. Because immersed submanifolds are the more general notion, many authors let
the unqualified word submanifold mean the immersed kind, with embedded reserved
for the special case. Others use submanifold for the embedded kind. To avoid the ambiguity
we always write embedded or immersed explicitly. Just as embedded submanifolds are
the images of embeddings, immersed submanifolds are the images of injective immersions.
Proposition: Images of Injective Immersions Are Submanifolds
Let \(N\) and \(M\) be smooth manifolds and let \(F : N \to M\) be an injective smooth
immersion. Then \(S = F(N)\) has a unique topology and smooth structure making it an immersed
submanifold of \(M\) for which \(F\) is a diffeomorphism onto \(S\).
Proof Sketch.
The construction parallels that for embeddings, except that the topology must now be
manufactured rather than inherited. We declare a set \(U \subseteq S\) open precisely when
\(F^{-1}(U)\) is open in \(N\). This makes \(F : N \to S\) a homeomorphism, so \(S\) is a
topological manifold. Transporting the charts of \(N\) across \(F\) gives a smooth structure for
which \(F\) is a diffeomorphism onto \(S\). The inclusion factors as \(\iota = F \circ F^{-1}\),
a diffeomorphism followed by the immersion \(F\), so it is a smooth immersion. This topology and
smooth structure are the only ones making \(F\) a diffeomorphism onto its image.
The two canonical examples are precisely the curves that failed to be embeddings earlier in the
manifold series. The
figure-eight curve and the
dense curve on the torus are images
of injective smooth immersions. As immersed submanifolds, each diffeomorphic to \(\mathbb{R}\), they
are perfectly well behaved. They are not embedded, because neither carries the subspace topology. At
the crossing point of the figure-eight, and everywhere along the dense torus curve, the ambient
neighborhoods cut the image into pieces that its own finer topology keeps together.
One can show, moreover, that no choice of topology and smooth structure can render their image sets
embedded. At the figure-eight's crossing point four half-branches meet, and no neighborhood of that
point is homeomorphic to an interval. For the dense torus curve the image in the subspace topology
is not even locally connected, so as a subspace it carries no manifold topology at all. The
obstruction is intrinsic to how the sets sit in the ambient space.
The failure to be embedded is, however, easily ruled out by any of the same global hypotheses that
upgraded immersions to embeddings.
Proposition: When an Immersed Submanifold Is Embedded
Let \(M\) be a smooth manifold and \(S \subseteq M\) an immersed submanifold. Then \(S\) is
embedded if any of the following holds:
(a) \(S\) has codimension \(0\).
(b) the inclusion \(S \hookrightarrow M\) is a proper map.
(c) \(S\) is compact.
Proof Sketch.
In every case the inclusion \(\iota : S \hookrightarrow M\) is already an injective immersion.
What must be supplied is that it is a topological embedding. The
sufficient conditions for an embedding
deliver exactly this from each hypothesis. Properness gives (b) directly. Compactness of \(S\)
makes \(\iota\) proper, since a continuous map from a compact space into the Hausdorff manifold
\(M\) is proper, and this yields (c). Codimension \(0\) means \(\dim S = \dim M\), so \(\iota\)
is an immersion between equidimensional manifolds, again an embedding. In each case \(\iota\) is
a smooth embedding, so \(S\) is embedded.
Each criterion echoes a fact from the embedded theory: codimension \(0\) forces openness, properness
is equivalent to closedness, and compactness implies properness. The same three conditions earlier
turned an injective immersion into an embedding, and here they are read off at the level of the
inclusion.
What an immersed submanifold always retains is the local structure of an embedded one.
Proposition: Immersed Submanifolds Are Locally Embedded
Let \(M\) be a smooth manifold and \(S \subseteq M\) an immersed submanifold. Then each point of
\(S\) has a neighborhood in \(S\) that is an embedded submanifold of \(M\).
Proof Sketch.
The inclusion \(\iota : S \hookrightarrow M\) is a smooth immersion, so by the
local embedding theorem
each \(p \in S\) has a neighborhood \(U\) in \(S\) on which \(\iota|_U\) is a smooth embedding.
Its image is then an embedded submanifold of \(M\).
It is essential to read this correctly. The proposition produces a neighborhood \(U\)
in \(S\) that is embedded. It does not claim a neighborhood \(V\) of \(p\) in
\(M\) for which \(V \cap S\) is embedded. For the dense torus curve no such \(V\) exists. Every
ambient neighborhood meets the curve in infinitely many strands. This gap between "a neighborhood in
\(S\)" and "the ambient trace of a neighborhood in \(M\)" is the exact difference between immersed
and embedded.
Parametrizations
For an immersed submanifold the inclusion need not be an embedding, so it is often more natural to
describe \(S\) by mapping into it from a Euclidean domain than by viewing it inside \(M\).
Definition: Local and Global Parametrizations
Let \(S \subseteq M\) be an immersed \(k\)-submanifold. A local parametrization
of \(S\) is a continuous map \(X : U \to M\), defined on an open set
\(U \subseteq \mathbb{R}^k\), whose image is an open subset of \(S\) and which, regarded as a
map into \(S\), is a homeomorphism onto its image. It is a
smooth local parametrization if, regarded as a map into \(S\), it is a
diffeomorphism onto its image. Such a map is in particular a homeomorphism, so every smooth
local parametrization is a local parametrization. If the image is all of \(S\), it is a
global parametrization.
Proposition: Parametrizations Are Inverse Charts
Let \(S \subseteq M\) be an immersed \(k\)-submanifold with inclusion \(\iota\), and let
\(U \subseteq \mathbb{R}^k\) be open. A map \(X : U \to M\) is a smooth local parametrization of
\(S\) if and only if there is a smooth chart \((V, \varphi)\) for \(S\) with
\(X = \iota \circ \varphi^{-1}\). In particular, every point of \(S\) lies in the image of some
smooth local parametrization.
Proof Sketch.
Charts for \(S\) are exactly the diffeomorphisms from open subsets of \(S\) onto open subsets
of \(\mathbb{R}^k\). One direction holds because a chart, read in its own coordinates, is the
identity map of an open subset of \(\mathbb{R}^k\) and so is smooth both ways. The other holds
because such a diffeomorphism is
smoothly compatible
with every chart of \(S\) and therefore belongs to the maximal atlas.
Suppose \(X : U \to M\) is a smooth local parametrization of \(S\). Its image \(V = X(U)\) is
open in \(S\), and \(X\), regarded as a map into \(S\), is a diffeomorphism onto \(V\). The
inverse \(\varphi = X^{-1} : V \to U\) therefore makes \((V, \varphi)\) a smooth chart, and
\(X = \iota \circ \varphi^{-1}\) by construction. Conversely, let \((V, \varphi)\) be a smooth
chart for \(S\) with \(U = \varphi(V)\). Then \(\varphi^{-1} : U \to V\) is a diffeomorphism
onto the open subset \(V\) of \(S\), so \(\iota \circ \varphi^{-1}\) is a smooth local
parametrization. The last claim holds because the charts of \(S\) cover \(S\).
A parametrization is thus nothing but a chart map read backwards, composed with the inclusion. The
most familiar instance is the graph. For a smooth function \(f : U \to \mathbb{R}^n\) on an open
\(U \subseteq \mathbb{R}^k\), the map \(\gamma_f(u) = (u, f(u))\) is a smooth global parametrization
of the graph \(\Gamma(f)\), inverse to the graph coordinate map. The open upper hemisphere of
\(\mathbb{S}^2\), parametrized by \((u, v) \mapsto (u, v, \sqrt{1 - u^2 - v^2})\), is the case at
hand.
The figure-eight curve, viewed as an
immersed submanifold of \(\mathbb{R}^2\), admits the very immersion that traces it as a smooth
global parametrization. A single chart presents the whole of it, despite its self-crossing in the
plane.
Submanifolds with Boundary
Every submanifold met so far has been a manifold without boundary, and so has every ambient
manifold. Integration is where that restriction becomes untenable. A region one integrates over is
a solid ball, a cylinder, or a plate, and what makes such a region interesting is exactly its edge.
For the region to carry an integral, and for that integral to be compared with one over the edge,
the region must be a manifold with boundary in its own right. It must also sit inside the ambient
manifold as faithfully as an embedded submanifold does. No new idea is needed. Each definition below is its
boundaryless counterpart with the word manifold replaced by manifold with boundary.
Definition: Submanifold with Boundary
Let \(M\) be a smooth manifold with or without boundary. A smooth submanifold with
boundary in \(M\) is a subset \(S \subseteq M\) endowed with a topology and a
smooth structure making it a
smooth manifold with boundary
for which the inclusion \(\iota : S \hookrightarrow M\) is a
smooth immersion.
If \(\iota\) is a
smooth embedding,
then \(S\) is an embedded submanifold with boundary, and in the general case
it is an immersed submanifold with boundary. The codimension
of \(S\) is \(\dim M - \dim S\), and \(S\) is properly embedded if it is
embedded and \(\iota\) is a
proper map.
Three readings of the definition prevent the most common confusions. First, the symbol
\(\partial S\) always denotes the manifold boundary of \(S\), the set of points that a chart of
\(S\) sends into \(\partial \mathbb{H}^n\). It is neither the topological boundary of the
subset \(S\) in \(M\) nor any part of \(\partial M\). A closed disk in the plane has the bounding
circle as its manifold boundary even though the plane has no boundary at all, and the
two meanings of the
symbol were separated when boundaries were first introduced.
Next, a manifold without boundary is a manifold with boundary whose boundary happens to be empty,
so every embedded submanifold in the earlier sense is an embedded submanifold with boundary, with
\(\partial S = \varnothing\). The new notion is a genuine extension rather than a competing one.
Finally, the terminological discipline of the previous section stays in force. The words embedded
and immersed are always written out, so that the unqualified phrase never has to be decoded.
Embedded submanifolds without boundary were characterized as the images of smooth embeddings. The
same characterization holds here, and it is proved by the same transport of structure.
Proposition: Images of Embeddings with Boundary
Let \(N\) be a smooth manifold with boundary, let \(M\) be a smooth manifold with or without
boundary, and let \(F : N \to M\) be a smooth embedding. Then \(S = F(N)\), with the
subspace topology, is a topological manifold with boundary, and it has a unique smooth
structure making it an embedded submanifold with boundary of \(M\) for which \(F\) is a
diffeomorphism onto \(S\). Its boundary is \(\partial S = F(\partial N)\).
Proof Sketch.
The argument given for the boundaryless case applies once charts modeled on \(\mathbb{H}^n\)
are admitted alongside those modeled on \(\mathbb{R}^n\). Because \(F\) is a topological
embedding, its corestriction \(F : N \to S\) is a homeomorphism, so \(S\) is a topological
manifold with boundary of the same dimension as \(N\). Declaring \((F(U), \varphi \circ
F^{-1})\) to be a chart of \(S\) for each chart \((U, \varphi)\) of \(N\) produces an
atlas whose transition maps are literally those of \(N\), so a transition read in the
half-space sense
remains smooth after the transport. With this atlas \(F : N \to S\) is a diffeomorphism, and
the inclusion factors as \(F\) after \(F^{-1}\), a diffeomorphism followed by an embedding,
hence is itself a smooth embedding. Uniqueness holds because any smooth structure for which
\(F\) is a diffeomorphism has exactly these charts, and \(\partial S = F(\partial N)\)
because a diffeomorphism carries boundary points to boundary points.
Two examples carry most of the weight later. The closed interval \([a, b]\) is a compact
\(1\)-manifold with boundary, so the image of any smooth embedding of it into a manifold is an
embedded \(1\)-dimensional submanifold with boundary, its boundary being the pair of endpoint
images. The closed unit ball \(\overline{\mathbb{B}}^n\), recorded earlier as a manifold with
boundary whose boundary is the unit sphere, sits in \(\mathbb{R}^n\) as an embedded submanifold
with boundary of codimension \(0\). Both are compact, and compactness settles properness for
free.
Proposition: Proper Embedding and Closedness with Boundary
An embedded submanifold with boundary \(S \subseteq M\) is properly embedded if and only if
it is a closed subset of \(M\). In particular, every compact embedded submanifold with
boundary is properly embedded.
Regular Domains
One class of submanifolds with boundary carries the whole weight of integration theory, and it is
the class of full-dimensional ones. A solid ball inside space, a plate inside the plane, and the
region under a graph are all of this kind, and each is closed in the ambient manifold.
Definition: Regular Domain
Let \(M\) be a smooth manifold with or without boundary. A regular domain in
\(M\) is a properly embedded
submanifold with boundary
of codimension \(0\).
The half-space \(\mathbb{H}^n\) inside \(\mathbb{R}^n\), the closed unit ball inside
\(\mathbb{R}^n\), and the closed upper hemisphere inside \(\mathbb{S}^n\) are regular domains.
None of them is open, and that is the entire point. Codimension \(0\) forced openness earlier on
this page, where submanifolds carried no boundary and an inclusion of full rank was therefore a
local diffeomorphism onto an open image. Admitting a boundary breaks the equivalence and admits
precisely the closed regions with an edge that a theory of integration over domains needs.
Because a regular domain is closed in \(M\) and has the full dimension, the two readings of the
symbol \(\partial\) that were carefully separated above collapse into one. This is the precise
form of the observation that the manifold boundary and the topological boundary agree whenever a
manifold sits in the ambient space as a closed subset of the same dimension.
Proposition: Interior and Boundary of a Regular Domain
Let \(M\) be a smooth manifold without boundary and let \(D \subseteq M\) be a regular
domain. Then \(\operatorname{Int} D = \operatorname{int}(D)\), the
topological interior
of \(D\) in \(M\), and \(\partial D = D \setminus \operatorname{int}(D)\), the
topological boundary. In particular \(\operatorname{Int} D\) is an open subset of \(M\).
Proof.
Write \(n = \dim M = \dim D\). The manifold interior \(\operatorname{Int} D\) is
an open subset of \(D\),
so it is an
open submanifold with boundary
of \(D\) whose boundary is \(\operatorname{Int} D \cap \partial D\). That intersection is empty,
because the
manifold interior and boundary are disjoint.
So \(\operatorname{Int} D\) is a smooth \(n\)-manifold without boundary, and its
inclusion into \(M\) is the restriction of a smooth immersion, hence a smooth immersion
between manifolds of equal dimension, neither of which has boundary. Such a map is a
local diffeomorphism
and therefore an
open map,
so \(\operatorname{Int} D\) is open in \(M\) and contained in the
topological interior
\(\operatorname{int}(D)\).
No point of \(\partial D\) lies in \(\operatorname{int}(D)\). Suppose \(p \in \partial
D\) had a neighborhood \(W\), open in \(M\), with \(W \subseteq D\). Since \(D\) is
embedded it carries the subspace topology, so \(W\) is open in \(D\) as well and the two
descriptions of \(W\) are descriptions of one topological space. Read inside \(D\), the set
\(W\) is an
open submanifold with boundary
with \(\partial W = W \cap \partial D\), which contains \(p\). Read inside \(M\), the
set \(W\) is an open subset of a manifold without boundary, so every chart of \(M\) presents
every point of \(W\) as an interior point. The point \(p\) would then be both an interior
point and a boundary point of \(W\), which the
invariance of the boundary
forbids.
The two statements now follow by set algebra. A regular domain is properly embedded, hence
closed in \(M\),
so its closure is \(D\) itself and its topological boundary is
\(D \setminus \operatorname{int}(D)\). The invariance of the boundary also gives
\(D = \operatorname{Int} D \sqcup \partial D\). The first paragraph puts
\(\operatorname{Int} D\) inside \(\operatorname{int}(D)\), and the second keeps
\(\partial D\) out of \(\operatorname{int}(D)\). Since \(\operatorname{int}(D)\) is
contained in \(D\) and misses \(\partial D\), it is contained in
\(\operatorname{Int} D\), so the two sets agree. Removing this set from \(D\) leaves
\(\partial D = D \setminus \operatorname{int}(D)\).
One supply of regular domains is worth recording, because arguments that test a pointwise
condition by integrating it need a small compact domain near an arbitrary point and need it inside
a prescribed open set.
Proposition: Compact Regular Domains from Coordinate Balls
Let \(M\) be a smooth \(n\)-manifold without boundary. If \(B \subseteq M\) is a
regular coordinate ball,
then \(\overline{B}\) is a compact regular domain in \(M\) with
\(\operatorname{Int} \overline{B} = B\). Consequently every nonempty open subset of \(M\)
contains a compact regular domain with nonempty interior.
Proof.
Let \(\varphi : B' \to B_{r'}(0)\) be the coordinate map supplied by the definition, with
\(\varphi(B) = B_r(0)\) and \(\varphi(\overline{B}) = \overline{B_r(0)}\). The closed
Euclidean ball \(\overline{B_r(0)}\) is the image of \(\overline{\mathbb{B}}^n\) under
the dilation \(x \mapsto rx\). By the example above it is therefore an embedded
codimension-\(0\) submanifold with boundary of \(\mathbb{R}^n\), and its inclusion into the
open set \(B_{r'}(0)\) is a smooth embedding. Composing that inclusion with the diffeomorphism
\(\varphi^{-1}\) gives a smooth embedding of \(\overline{B_r(0)}\) into \(B'\) with
image \(\overline{B}\), so \(\overline{B}\) is an
embedded submanifold with boundary
of \(B'\) of codimension \(0\). Since \(B'\) is open in \(M\), the inclusion of
\(\overline{B}\) into \(M\) is again a smooth embedding. The set \(\overline{B}\) is
compact, being the image of the compact set \(\overline{B_r(0)}\) under the homeomorphism
\(\varphi^{-1}\), so it is closed and therefore
properly embedded.
It is thus a regular domain,
and its manifold interior corresponds under \(\varphi\) to the open ball \(B_r(0)\), that
is, to \(B\).
For the last claim let \(U \subseteq M\) be open and nonempty and pick \(p \in U\). The
regular coordinate balls form a basis,
so there is one, say \(B_0\) with coordinate map \(\varphi_0\) and radius \(r_0\),
satisfying \(p \in B_0 \subseteq U\). Set
\(B_1 = \varphi_0^{-1}\bigl(B_{r_0/2}(0)\bigr)\). Because \(\varphi_0\) is a
homeomorphism onto its image, the set \(\varphi_0^{-1}\bigl(\overline{B_{r_0/2}(0)}\bigr)\)
is the closure of \(B_1\) in \(B_0\), and being compact it is closed in \(M\), so it is
also the closure \(\overline{B_1}\) taken in \(M\). The triple consisting of \(B_1\), its
closure, and \(B_0\) satisfies the radius conditions with \(r_0/2 \lt r_0\), so \(B_1\)
is itself a regular coordinate ball. Its closure is therefore a compact regular domain with
interior \(B_1\), and it lies in \(B_0\), hence in \(U\).
The language of embedded, level set, and immersed submanifolds is now in place, together with the
regions with an edge that integration will demand. We have the vocabulary to ask the two
technical questions that the rest of the theory answers: when a smooth map can be restricted to a
submanifold without losing smoothness, and how the tangent space to a submanifold sits inside the
tangent space of its ambient manifold.