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 — one
that 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 —
that its inclusion 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: it must be the subspace topology inherited from \(M\). 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; throughout this page the ambient manifold
\(M\) is likewise taken without boundary, so that every map in sight is of the interior kind. 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; and 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 \(M\). Transport the smooth
structure across this homeomorphism: declare 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; so this
smooth structure makes \(S\) an embedded submanifold. Uniqueness holds because any smooth
structure on \(S\) for which \(F\) is a diffeomorphism must have exactly these charts.
\(\blacksquare\)
This proposition is the working characterization of embedded submanifolds: they 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, beginning 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. \(\blacksquare\)
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: \(\gamma_f\) 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 \(U\).
\(\blacksquare\)
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 — restricting and extending smooth
functions among them — require a submanifold to be closed as a subset, and this turns out to be
equivalent to a clean condition on the inclusion map.
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 — a distinction
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.
\(\blacksquare\)
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 — the way a single horizontal plane \(\{z = c\}\) sits in
\(\mathbb{R}^3\). 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 \(k\), so the
rank theorem
supplies a chart \((V_0, \psi)\) 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 \(\psi\) 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; so the inclusion is a smooth embedding and \(S\) is
an embedded \(k\)-submanifold. That this structure is the only one making \(S\) a
submanifold — that no other topology or smooth structure on \(S\) renders the inclusion an
embedding or even an immersion — is a separate matter, resting on the global rank theorem; it is
established once we have the tools to restrict maps to submanifolds, where the
uniqueness
is proved in full. \(\blacksquare\)
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
\(\{x : x^i > 0\}\) in the graph of the smooth function
\(x^i = \sqrt{1 - \sum_{j \neq i} (x^j)^2}\), and each \(\{x : x^i < 0\}\) 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 — and 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 \(\mathbb{R}^2 \to \mathbb{R}\),
\[
\Theta(x, y) = x^2 - y, \qquad \Phi(x, y) = x^2 - y^2, \qquad \Psi(x, y) = x^2 - y^3,
\]
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; but 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 — and it is supplied by the rank theorem.
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 \(m = \dim M\), \(n = \dim N\), 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 slice \(\{x^1 = \dots = x^r = 0\}\), so \((U, \varphi)\)
is 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.
\(\blacksquare\)
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 \(r = \dim N\).
\(\blacksquare\)
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 — its differential is surjective at every point —
and each of its \(r\) local component functions likewise removes one dimension, leaving 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; otherwise — that is, when
\(\Phi^{-1}(c)\) contains at least one critical point — it 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.)
\(\blacksquare\)
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 trading 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, and if it is real- or vector-valued, 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 \(U\), so \(S\) satisfies the local slice
condition and is an embedded submanifold of \(M\). \(\blacksquare\)
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
\(\{(r, z) : r > 0\}\) cut out locally by \(\varphi(r, z) = 0\), and revolve it about the
\(z\)-axis. The resulting surface
\(S_C = \{(x, y, z) : \varphi(\sqrt{x^2 + y^2},\, z) = 0\}\) is then the level set of
\(\Phi(x, y, z) = \varphi(\sqrt{x^2 + y^2},\, z)\), 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 \(d\varphi\) is; thus where
\(\varphi\) defines \(C\) regularly, \(\Phi\) defines \(S_C\) regularly, exhibiting
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 — images, sensor readings, the activations of a network — rarely
fill their ambient space \(\mathbb{R}^n\); they cluster 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 \(r\). The intuition that data occupies a
low-dimensional surface inside a high-dimensional space — and that learning a representation
means recovering that surface — is, in this language, the statement that the data lies on or
near an embedded submanifold of \(\mathbb{R}^n\). The
manifold viewpoint on data
thus rests on exactly the constructions of this page; what remains — that such a submanifold can
always be situated inside a Euclidean space of controlled dimension — is the embedding theory
the next stage of the manifold series develops.
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, the image of a
curve that returns arbitrarily close to itself. 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: 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. \(\blacksquare\)
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, no neighborhood
of which is homeomorphic to an interval; and 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 — a continuous map from a compact space into the Hausdorff manifold \(M\)
is proper — yielding (c); and 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. \(\blacksquare\)
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 that
earlier turned an injective immersion into an embedding, now 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 \(M\). \(\blacksquare\)
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 — in particular then 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.
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 \(f : U \to \mathbb{R}^k\) on an open
\(U \subseteq \mathbb{R}^n\), 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. And 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 presenting the whole of it, despite its self-crossing
in the plane. With the language of embedded, level set, and immersed submanifolds now in place, 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.