Approximating Continuous Functions
The embedding theorem placed every manifold inside a Euclidean space. The second application of
Sard's circle of ideas concerns approximation. A continuous map can always be deformed into a
smooth one, and a continuous map between manifolds is homotopic to a smooth one. The first claim,
for functions valued in a Euclidean space, is elementary and uses only a partition of unity. We
prove it here, and it becomes the analytic core of everything that follows. The passage to maps
between manifolds is deferred to the final section, because it requires the tubular neighborhood,
a geometric construction built in between.
The right notion of approximation allows the tolerance to vary from point to point, which is what
makes the statement strong enough to be useful on noncompact manifolds. Given a positive
continuous function \(\delta : M \to \mathbb{R}\), two maps
\(F, \widetilde{F} : M \to \mathbb{R}^k\) are \(\delta\)-close if
\(\lvert F(x) - \widetilde{F}(x) \rvert \lt \delta(x)\) for every \(x \in M\). A uniform tolerance
is the special case of a constant \(\delta\). Letting \(\delta\) shrink where the manifold runs
off to its ends makes the approximation as tight as we like there.
Theorem (Whitney Approximation Theorem for Functions)
Let \(M\) be a smooth manifold with or without boundary, and let \(F : M \to \mathbb{R}^k\) be
a continuous function. For any positive continuous function \(\delta : M \to \mathbb{R}\),
there is a smooth function \(\widetilde{F} : M \to \mathbb{R}^k\) that is \(\delta\)-close to
\(F\). If \(F\) is already smooth on a closed subset \(A \subseteq M\), then \(\widetilde{F}\)
can be chosen to equal \(F\) on \(A\).
Proof.
If \(F\) is smooth on the closed set \(A\), the
extension lemma
provides a smooth function \(F_0 : M \to \mathbb{R}^k\) agreeing with \(F\) on \(A\). Let
\[
U_0 = \{\, y \in M : \lvert F_0(y) - F(y) \rvert \lt \delta(y) \,\},
\]
an open set containing \(A\), since \(F_0\) and \(F\) agree there. (If there is no such \(A\),
take \(U_0 = A = \varnothing\) and \(F_0 \equiv 0\).)
Away from \(A\) we approximate locally by constants and patch. We claim there are countably
many points \(\{x_i\}\) in \(M \setminus A\) and neighborhoods \(U_i\) of \(x_i\) covering
\(M \setminus A\), such that
\[
\lvert F(y) - F(x_i) \rvert \lt \delta(y) \quad \forall y \in U_i.
\]
Indeed, around any \(x \in M \setminus A\), continuity of \(F\) and \(\delta\) lets us choose
a neighborhood \(U_x \subseteq M \setminus A\) small enough that
\(\delta(y) \gt \tfrac12 \delta(x)\) and \(\lvert F(y) - F(x) \rvert \lt \tfrac12 \delta(x)\)
hold on it. Then for \(y \in U_x\),
\[
\lvert F(y) - F(x) \rvert \lt \tfrac12 \delta(x) \lt \delta(y).
\]
The collection \(\{U_x\}\) covers \(M \setminus A\), which is an
open submanifold
of \(M\) and hence second countable in its own right, so the cover has a
countable subcover.
That subcover gives the \(\{x_i\}\) and \(\{U_i\}\) with the displayed property.
Now take a smooth
partition of unity
\(\{\varphi_0, \varphi_i\}\) subordinate to the open cover \(\{U_0, U_i\}\) of \(M\). These
sets do cover \(M\), since \(U_0 \supseteq A\) and the \(U_i\) cover \(M \setminus A\). Define
\[
\widetilde{F}(y) = \varphi_0(y) F_0(y) + \sum_{i \ge 1} \varphi_i(y)\, F(x_i).
\]
Smoothness of \(\widetilde{F}\) follows because \(F_0\) is smooth, the values \(F(x_i)\) are
constants, and the partition of unity is smooth and locally finite. On \(A\) every
\(\varphi_i\) with \(i \ge 1\) vanishes, since its support misses \(A\). Hence
\(\widetilde{F} = \varphi_0 F_0 = F_0 = F\) there.
Finally we estimate the error. Since \(\sum_{i \ge 0} \varphi_i \equiv 1\), we may write
\(F(y) = \big(\varphi_0(y) + \sum_{i \ge 1} \varphi_i(y)\big) F(y)\) and subtract:
\[
\big\lvert \widetilde{F}(y) - F(y) \big\rvert
= \Big\lvert \varphi_0(y)\big(F_0(y) - F(y)\big)
+ \sum_{i \ge 1} \varphi_i(y)\big(F(x_i) - F(y)\big) \Big\rvert .
\]
Each surviving term is controlled. Where \(\varphi_0(y) \ne 0\) we have \(y \in U_0\), so
\(\lvert F_0(y) - F(y) \rvert \lt \delta(y)\). Where \(\varphi_i(y) \ne 0\) we have
\(y \in U_i\), so \(\lvert F(x_i) - F(y) \rvert \lt \delta(y)\) by the local estimate above.
The triangle inequality and \(\sum_{i \ge 0} \varphi_i(y) = 1\) give
\[
\big\lvert \widetilde{F}(y) - F(y) \big\rvert
\lt \Big( \varphi_0(y) + \sum_{i \ge 1} \varphi_i(y) \Big)\, \delta(y) = \delta(y),
\]
so \(\widetilde{F}\) is \(\delta\)-close to \(F\).
A small reformulation of the theorem is worth recording on its own. A positive continuous function
can always be underestimated by a positive smooth one. That reformulation is the device
that converts a merely continuous tolerance into one we can differentiate.
Corollary: A Smaller Positive Smooth Function
If \(M\) is a smooth manifold with or without boundary and \(\delta : M \to \mathbb{R}\) is a
positive continuous function, then there is a positive smooth function \(e : M \to
\mathbb{R}\) with \(0 \lt e(x) \lt \delta(x)\) for all \(x \in M\).
Proof.
Applying the theorem to the continuous function \(\tfrac12 \delta\) with tolerance
\(\tfrac12 \delta\) produces a smooth \(e\) with
\(\lvert e(x) - \tfrac12 \delta(x) \rvert \lt \tfrac12 \delta(x)\) for all \(x\), and this
inequality is exactly \(0 \lt e(x) \lt \delta(x)\).
The Normal Bundle and Tubular Neighborhoods
The approximation theorem just proved smooths out functions valued in a Euclidean space. To smooth
a continuous map valued in a manifold \(M\), we will first embed \(M\) in some \(\mathbb{R}^n\),
approximate in \(\mathbb{R}^n\), and then push the approximation back onto \(M\). The last step
needs a smooth map that sends nearby points of \(\mathbb{R}^n\) back to \(M\). Such a map is a
retraction of a neighborhood of \(M\) onto \(M\). Constructing it is the geometric heart of this
section, and the construction runs through the normal directions to \(M\).
Throughout, \(M \subseteq \mathbb{R}^n\) is an embedded \(m\)-dimensional submanifold without
boundary. The identification of \(T_x\mathbb{R}^n\) with \(\mathbb{R}^n\) carries the Euclidean
dot product onto each tangent space, and we use it to speak of orthogonality.
Definition: Normal Space and Normal Bundle
For \(x \in M\), the normal space to \(M\) at \(x\) is the orthogonal
complement of the
tangent space
inside \(T_x\mathbb{R}^n \cong \mathbb{R}^n\),
\[
N_x M = (T_x M)^{\perp} \subseteq \mathbb{R}^n,
\]
an \((n - m)\)-dimensional subspace.
The normal bundle of \(M\) is the set of all normal vectors at all points,
\[
NM = \{\, (x, v) \in \mathbb{R}^n \times \mathbb{R}^n : x \in M,\ v \in N_x M \,\}
\subseteq T\mathbb{R}^n,
\]
viewed as a subset of
the tangent bundle
\(T\mathbb{R}^n \cong \mathbb{R}^n \times \mathbb{R}^n\), with natural projection
\(\pi_{NM} : NM \to M\) sending \((x, v)\) to \(x\).
The normal bundle is itself a smooth manifold, and of exactly the ambient dimension. The \(m\)
directions along \(M\) and the \(n - m\) normal directions together restore all \(n\). That
dimension count is what lets us treat \(NM\) like the domain of a chart.
Theorem: The Normal Bundle Is an Embedded Submanifold
If \(M \subseteq \mathbb{R}^n\) is an embedded \(m\)-dimensional submanifold without boundary,
then \(NM\) is an embedded \(n\)-dimensional submanifold of
\(T\mathbb{R}^n \cong \mathbb{R}^n \times \mathbb{R}^n\).
Proof.
Fix \(x_0 \in M\) and a
slice chart
\((U, \varphi)\) for \(M\) centered at \(x_0\), with coordinate functions
\((u^1, \dots, u^n)\) so that \(M \cap U\) is cut out by \(u^{m+1} = \dots = u^n = 0\). At
each point \(x \in U\), the vectors
\[
E_j\big|_x = (d\varphi_x)^{-1}\Big(\tfrac{\partial}{\partial u^j}\Big|_{\varphi(x)}\Big),
\quad j = 1, \dots, n,
\]
form a basis for \(T_x\mathbb{R}^n\). Writing \(E_j|_x = E_j^i(x)\, \partial/\partial x^i\),
each component \(E_j^i\) is a partial derivative of \(\varphi^{-1}\), hence a smooth function
of \(x\). For \(x \in M \cap U\), the first \(m\) of these vectors span \(T_x M\).
Define
\[
\Phi : U \times \mathbb{R}^n \to \widehat{U} \times \mathbb{R}^n,
\quad
\Phi(x, v) = \big(u^1(x), \dots, u^n(x),\, v \cdot E_1|_x, \dots, v \cdot E_n|_x\big),
\]
where \(\widehat{U} = \varphi(U)\). Its total derivative at \((x, v)\) has block form
\[
D\Phi_{(x, v)} =
\begin{pmatrix}
\dfrac{\partial u^i}{\partial x^j}(x) & 0 \\\\
* & E_i^j(x)
\end{pmatrix},
\]
whose diagonal blocks are invertible. The upper block is invertible because \(\varphi\) is a
diffeomorphism, and the lower because the \(E_j\) form a basis, so \(\Phi\) is a local
diffeomorphism.
The map \(\Phi\) is also injective on \(U \times \mathbb{R}^n\). If
\(\Phi(x, v) = \Phi(x', v')\), the first \(n\) coordinates force \(x = x'\) since \(\varphi\)
is injective, and then \(v \cdot E_i|_x = v' \cdot E_i|_x\) for all \(i\) forces \(v = v'\)
since the \(E_i\) are a basis. Thus \(\Phi\) is a diffeomorphism onto its image, a smooth
coordinate chart on \(U \times \mathbb{R}^n\).
In these coordinates \(NM\) is a slice. A pair \((x, v)\) lies in \(NM\) exactly when
\(x \in M \cap U\), so \(u^{m+1}(x) = \dots = u^n(x) = 0\), and \(v \perp T_x M\), so the
components \(v \cdot E_1|_x = \dots = v \cdot E_m|_x = 0\). These are \(n\) coordinate
functions of \(\Phi\) set to zero, exhibiting \(NM\) locally as the slice where they vanish.
Hence \(\Phi\) is a slice chart for \(NM\), and since such charts exist around every point,
\(NM\) is an embedded \(n\)-dimensional submanifold.
The normal bundle organizes the directions in which one leaves \(M\). Adding a normal vector to its
base point pushes off \(M\) along that direction, and doing so for all small normal vectors sweeps
out a neighborhood of \(M\). The map that performs this is addition.
Definition: Tubular Neighborhood
Let \(E : NM \to \mathbb{R}^n\) be the smooth map \(E(x, v) = x + v\), the restriction to
\(NM\) of the addition map \(\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n\). A
tubular neighborhood of \(M\) is a neighborhood \(U\) of \(M\) in
\(\mathbb{R}^n\) that is the diffeomorphic image under \(E\) of an open subset \(V \subseteq
NM\) of the form
\[
V = \{\, (x, v) \in NM : \lvert v \rvert \lt \delta(x) \,\}
\]
for some positive continuous function \(\delta : M \to \mathbb{R}\).
A tubular neighborhood is a thickening of \(M\) by a variable radius, fibered by the normal disks.
Its existence is the central construction of the section.
Theorem (Tubular Neighborhood Theorem)
Every embedded submanifold of \(\mathbb{R}^n\) without boundary has a tubular neighborhood.
Proof.
Let \(M_0 = \{(x, 0) : x \in M\} \subseteq NM\) be the zero section. We first show that \(E\)
is a local diffeomorphism near every point of \(M_0\). By the
inverse function theorem for manifolds,
it suffices that \(dE_{(x, 0)}\) be bijective.
Two restrictions of \(E\) reveal its differential. Restricted to the zero section, \(E\) is
the obvious diffeomorphism \(M_0 \to M\), so \(dE_{(x,0)}\) carries \(T_{(x,0)} M_0\)
isomorphically onto \(T_x M\). Restricted to the fiber \(N_x M\), the map \(E\) is the affine
map \(w \mapsto x + w\), so \(dE_{(x,0)}\) carries \(T_{(x,0)}(N_x M)\) isomorphically onto
\(N_x M\). Since \(T_x\mathbb{R}^n = T_x M \oplus N_x M\), the differential \(dE_{(x,0)}\) is
surjective, hence bijective by equality of dimensions.
So \(E\) restricts to a diffeomorphism on a neighborhood of \((x, 0)\). Since \(NM\) is
embedded in \(\mathbb{R}^n \times \mathbb{R}^n\) and carries the Euclidean metric, we may take
that neighborhood of the form
\(V_\delta(x) = \{(x', v') \in NM : \lvert x - x' \rvert \lt \delta,\ \lvert v' \rvert \lt \delta\}\)
for some \(\delta \gt 0\).
It remains to find a single open set of the form
\(V = \{(x, v) : \lvert v \rvert \lt \delta(x)\}\) on which \(E\) is a global diffeomorphism.
For each \(x \in M\), let \(\rho(x)\) be the supremum of all \(\delta \le 1\) for which \(E\)
is a diffeomorphism from \(V_\delta(x)\) onto its image. The previous paragraph makes
\(\rho(x)\) positive.
For two nearby base points \(x, x'\), a ball about \(x'\) of radius \(\delta\) is contained in
one about \(x\) of radius \(\delta + \lvert x - x' \rvert\). Comparing the neighborhoods
around \(x\) and \(x'\) therefore gives the estimate
\(\rho(x) - \rho(x') \le \lvert x - x' \rvert\), and by symmetry \(\rho\) is continuous
(indeed \(1\)-Lipschitz). Set
\(V = \{(x, v) \in NM : \lvert v \rvert \lt \tfrac12 \rho(x)\}\), an open set of the required
form with the positive continuous radius function \(\tfrac12\rho\).
On \(V\) the map \(E\) is injective. Suppose \(E(x, v) = E(x', v')\) with both points in
\(V\), and say \(\rho(x') \le \rho(x)\). From \(x + v = x' + v'\) we get \(x - x' = v' - v\),
so
\[
\lvert x - x' \rvert = \lvert v' - v \rvert \le \lvert v \rvert + \lvert v' \rvert
\lt \tfrac12\rho(x) + \tfrac12\rho(x') \le \rho(x),
\]
using \(\lvert v \rvert \lt \tfrac12\rho(x)\), \(\lvert v' \rvert \lt \tfrac12\rho(x')\), and
\(\rho(x') \le \rho(x)\). Together with \(\lvert v \rvert, \lvert v' \rvert \lt \rho(x)\),
this places both \((x, v)\) and \((x', v')\) inside \(V_{\rho(x)}(x)\). The sets
\(V_\delta(x)\) increase with \(\delta\), and for every \(\delta \lt \rho(x)\) the supremum
defining \(\rho(x)\) supplies a larger radius at which \(E\) is a diffeomorphism. So \(E\)
restricts to a diffeomorphism on \(V_\delta(x)\) for every \(\delta \lt \rho(x)\), and any two
points of the union \(V_{\rho(x)}(x)\) already lie in a common such \(V_\delta(x)\). Hence
\((x, v) = (x', v')\).
Every point \((y, w) \in V\) satisfies \(\lvert w \rvert \lt \tfrac12 \rho(y)\) and so lies in
\(V_\delta(y)\) for any \(\delta\) between \(\tfrac12\rho(y)\) and \(\rho(y)\), which by the
previous paragraph makes \(E\) a local diffeomorphism at every point of \(V\). Local
diffeomorphisms are
open maps, and bijective ones are diffeomorphisms,
so \(U = E(V)\) is open and the injective map \(E : V \to U\) is a diffeomorphism. Thus \(U\)
is a tubular neighborhood of \(M\), with radius function \(\tfrac12\rho\).
The payoff is the retraction we set out to build. A tubular neighborhood comes with a canonical
map back onto \(M\). We undo the diffeomorphism \(E\) and forget the normal vector.
Proposition: Tubular Neighborhoods Retract onto the Submanifold
Let \(M \subseteq \mathbb{R}^n\) be an embedded submanifold without boundary, with tubular
neighborhood \(U\). Then there is a smooth map \(r : U \to M\) that is both a
retraction, meaning that \(r|_M\) is the identity, and a
submersion.
Proof.
Write \(U = E(V)\) with \(E : V \to U\) a diffeomorphism, and set
\(r = \pi_{NM} \circ E^{-1}\), the composition of the inverse diffeomorphism with the bundle
projection \(\pi_{NM} : NM \to M\). It is smooth as a composition of smooth maps. For
\(x \in M\) we have \(E^{-1}(x) = (x, 0)\), so \(r(x) = \pi_{NM}(x, 0) = x\), making \(r\) a
retraction.
To see that \(r\) is a submersion it is enough that \(\pi_{NM}\) be one, since \(E^{-1}\) is a
diffeomorphism. In the slice chart \(\Phi\) built above, \(NM\) carries coordinates
\((u^1, \dots, u^m)\) along the base together with the normal-fiber coordinates, and
\(\pi_{NM}\) reads off the first \(m\) of them while \(M\) carries \((u^1, \dots, u^m)\) as
its chart. In these coordinates \(\pi_{NM}\) is the projection
\((u^1, \dots, u^m, \text{fiber}) \mapsto (u^1, \dots, u^m)\), whose differential is
surjective. Hence \(\pi_{NM}\), and therefore \(r\), is a submersion.
Approximating Maps Between Manifolds
Now the two halves combine. To smooth a continuous map \(F : N \to M\) into a manifold, embed
\(M\) in \(\mathbb{R}^n\) by the Whitney theorem, approximate \(F\) by a smooth map into
\(\mathbb{R}^n\) using the approximation theorem for functions, and retract the result back onto
\(M\) through the tubular neighborhood. The retraction is what keeps the smoothed map valued in
\(M\) rather than drifting into the ambient space. The straight-line path from \(F\) to its
approximation, pushed down onto \(M\), supplies a homotopy for free.
Theorem (Whitney Approximation Theorem)
Let \(N\) be a smooth manifold with or without boundary, let \(M\) be a smooth manifold
without boundary, and let \(F : N \to M\) be a continuous map. Then \(F\) is homotopic to a
smooth map. If \(F\) is already smooth on a closed subset \(A \subseteq N\), the homotopy can
be taken relative to \(A\).
Proof.
By the
Whitney embedding theorem
we may regard \(M\) as a properly embedded submanifold of some \(\mathbb{R}^n\). Let \(U\) be
a tubular neighborhood
of \(M\) and \(r : U \to M\) the associated
smooth retraction. For
\(x \in M\) set
\[
\delta(x) = \sup\{\, \varepsilon \le 1 : B_\varepsilon(x) \subseteq U \,\},
\]
a positive function, continuous by a triangle-inequality argument like the one for the tubular
neighborhood theorem.
Then \(\widetilde\delta = \delta \circ F : N \to \mathbb{R}\) is positive and continuous, so
by the
approximation theorem for functions
there is a smooth map \(\widetilde{F} : N \to \mathbb{R}^n\) that is
\(\widetilde\delta\)-close to \(F\) and equal to \(F\) on \(A\).
Define \(H : N \times I \to M\) by
\[
H(p, t) = r\big((1 - t) F(p) + t \widetilde{F}(p)\big).
\]
The map \(H\) is well defined. For each \(p\), the bound
\(\lvert \widetilde{F}(p) - F(p) \rvert \lt \widetilde\delta(p) = \delta(F(p))\) means that
\(\widetilde{F}(p)\) lies in the ball \(B_{\delta(F(p))}(F(p)) \subseteq U\), and since that
ball is convex the entire segment from \(F(p)\) to \(\widetilde{F}(p)\) lies in \(U\), where
\(r\) is defined.
Then \(H(p, 0) = r(F(p)) = F(p)\), because \(F(p) \in M\) and \(r\) is a retraction, while
\(H(p, 1) = r(\widetilde{F}(p))\) is smooth in \(p\) as a composition of smooth maps. Thus
\(H\) is a homotopy from \(F\) to the smooth map \(r \circ \widetilde{F}\). On \(A\) we have
\(\widetilde{F} = F\), so the segment is constant and \(H(p, t) = r(F(p)) = F(p)\) throughout.
The homotopy is therefore relative to \(A\).
Replacing the codomain \(\mathbb{R}^k\) of the function theorem by a manifold turns approximation
into homotopy, and it also upgrades the extension lemma. A continuous map that is smooth on a
closed set, and that extends continuously, extends smoothly.
Corollary (Extension Lemma for Smooth Maps)
Let \(N\) be a smooth manifold with or without boundary, \(M\) a smooth manifold without
boundary, and \(A \subseteq N\) a closed subset. A smooth map \(f : A \to M\) has a smooth
extension to \(N\) if and only if it has a continuous extension to \(N\).
Proof.
A smooth extension is in particular continuous, so one direction is immediate. Conversely, if
\(F : N \to M\) is a continuous extension of \(f\), then \(F\) is smooth on the closed set
\(A\), so the approximation theorem produces a homotopy relative to \(A\) from \(F\) to a
smooth map \(\widetilde{F}\). Being relative to \(A\), this homotopy fixes \(A\), so
\(\widetilde{F}\) agrees with \(F\), hence with \(f\), on \(A\). The map \(\widetilde{F}\) is
therefore a smooth extension of \(f\).
The approximation theorem also lets us upgrade homotopies themselves from continuous to smooth, a
refinement used whenever a homotopy-theoretic argument needs to be carried out with calculus. A
homotopy \(H : N \times I \to M\) is a smooth homotopy if it extends to a smooth
map on a neighborhood of \(N \times I\) in \(N \times \mathbb{R}\). Two maps are smoothly
homotopic when such a homotopy connects them.
Lemma: Smooth Homotopy Is an Equivalence Relation
Let \(N\) be a smooth manifold with or without boundary and \(M\) a smooth manifold without
boundary. On the set of smooth maps from \(N\) to \(M\), smooth homotopy is an equivalence
relation.
Proof.
Reflexivity and symmetry are immediate. For transitivity, suppose \(H_1\) is a smooth homotopy
from \(F\) to \(G\) and \(H_2\) from \(G\) to \(K\). Naively concatenating them at
\(t = \tfrac12\) would generally produce only a continuous map, with a corner in the time
variable. We remove the corner by reparametrizing time with a smooth nondecreasing function
\(\varphi : [0, 1] \to [0, 2]\) that is \(0\) near \(0\), is \(2\) near \(1\), and is
identically \(1\) on a neighborhood of \(\tfrac12\). Monotonicity keeps
\(\varphi([0, \tfrac12]) \subseteq [0, 1]\) and \(\varphi([\tfrac12, 1]) \subseteq [1, 2]\),
so both pieces below are defined. Define
\[
H(x, t) =
\begin{cases}
H_1\big(x, \varphi(t)\big), & t \in [0, \tfrac12], \\\\
H_2\big(x, \varphi(t) - 1\big), & t \in [\tfrac12, 1].
\end{cases}
\]
Near \(t = \tfrac12\) both pieces equal the constant-in-\(t\) map \(G\) (since
\(\varphi \equiv 1\) there), so they agree to all orders. Hence \(H\) is smooth, and it is a
smooth homotopy from \(F\) to \(K\).
Finally, the distinction between continuous and smooth homotopy collapses for smooth maps. Any two
smoothly-mappable endpoints that are homotopic at all are smoothly homotopic.
Theorem: Homotopic Smooth Maps Are Smoothly Homotopic
Let \(N\) be a smooth manifold with or without boundary, \(M\) a smooth manifold without
boundary, and \(F, G : N \to M\) smooth maps. If \(F\) and \(G\) are homotopic, they are
smoothly homotopic. If they are homotopic relative to a closed subset \(A \subseteq N\), they
are smoothly homotopic relative to \(A\).
Proof.
Let \(H : N \times I \to M\) be a homotopy from \(F\) to \(G\). Extend it to
\(N \times \mathbb{R}\) by holding it constant outside \([0, 1]\) in the time variable,
setting \(\overline{H}(x, t) = H(x, 0)\) for \(t \le 0\) and \(\overline{H}(x, t) = H(x, 1)\)
for \(t \ge 1\). This \(\overline{H}\) is continuous by the closed-set form of the
gluing lemma for continuous maps,
applied to \(N \times (-\infty, 0]\), \(N \times I\) and \(N \times [1, \infty)\), and it is
already smooth on the closed set \(N \times \{0\} \cup N \times \{1\}\), where it equals the
smooth maps \(F\) and \(G\).
Since \(N \times \mathbb{R}\) is again a smooth manifold with or without boundary, the
approximation theorem provides a smooth map agreeing with \(\overline{H}\) on that closed set.
Restricting it to \(N \times I\) gives a smooth homotopy from \(F\) to \(G\). The relative
version follows by carrying the subset \(A\) through the same construction.
A word is in order on the hypothesis that \(M\) has no boundary, which has been standing since the
approximation theorem. The retraction onto \(M\) is built from a tubular neighborhood, and a
manifold with boundary has no two-sided collar of normal directions along its boundary, so the
construction breaks down there. The analogues of these theorems for a target with boundary require
different tools.
Embed, surround, retract: the manifold hypothesis completed
These results close a circle. The
embedding theorem
placed an abstract data manifold inside a Euclidean space. The tubular neighborhood theorem
surrounds that manifold with a tube of ambient points. The retraction \(r : U \to M\) sends
every point of the tube back to the base point of the normal fiber through it.
This map is the geometric prototype of denoising. A noisy observation, lying off the data
manifold but within its tube, is carried back along its normal direction to a clean point on
the manifold. A learned denoiser estimates a probabilistic version of this, a conditional
expectation rather than an exact projection. The retraction still captures the geometric
essence of denoising, the collapse of the normal directions. The
manifold hypothesis
thereby acquires not just a geometry, data on a low-dimensional surface in a high-dimensional
space, but a dynamics. We embed the manifold, surround it with a tube, and
retract noise onto it.
The normal bundle that carries the construction is a basic example of a vector bundle, the
kind of structure on which the manifold series builds its later geometry. The normal
directions it organizes are exactly the directions a model perturbs when it moves a point off
a learned manifold or projects one back onto it.