Bundle Homomorphisms
A
vector bundle
is a space equipped with a fiber structure (a smooth manifold, in the smooth case), so the natural
maps between vector bundles must respect both the structure of the total space (continuity or
smoothness) and the fiber structure (linearity, base point assignment). The result is the class of
bundle homomorphisms, which descend to a map of base spaces and act linearly on each fiber. These
are the morphisms in the category of vector bundles, and they are the right tool for discussing
subbundles, quotient bundles, and the differential of a smooth map.
Definition: Bundle Homomorphism
Let \(\pi : E \to M\) and \(\pi' : E' \to M'\) be vector bundles. A
bundle homomorphism from \(E\) to \(E'\) is a continuous map \(F : E \to E'\)
for which there exists a map \(f : M \to M'\) satisfying \(\pi' \circ F = f \circ \pi\), with
the property that the restriction \(F|_{E_p} : E_p \to E'_{f(p)}\) is linear for each
\(p \in M\). One says that \(F\) covers \(f\), and that \(f\) is the
base map of \(F\). When the bundles and \(F\) are smooth, \(F\) is called a
smooth bundle homomorphism.
The base map \(f\) is uniquely determined by \(F\). The relation
\(\pi' \circ F = f \circ \pi\) forces \(f(\pi(v)) = \pi'(F(v))\) for every \(v \in E\), and the
surjectivity of \(\pi\) means this determines \(f\) at every point of \(M\). The next proposition records that the regularity of \(f\)
is automatic.
Proposition: The Base Map Is Determined and Inherits Regularity
Let \(F : E \to E'\) be a bundle homomorphism covering \(f : M \to M'\). Then \(f\) is
continuous and uniquely determined by \(F\). If the bundles and \(F\) are smooth, \(f\) is
smooth.
Proof:
Uniqueness was noted in the discussion above. For regularity, observe that
\(f = \pi' \circ F \circ \zeta\), where \(\zeta : M \to E\) is the
zero section
of \(E\). Indeed, \(\zeta(p) \in E_p\), so \(F(\zeta(p)) \in E'_{f(p)}\), and applying \(\pi'\)
gives \(f(p)\). Continuity of \(f\) follows because \(\zeta\), \(F\), and \(\pi'\) are
continuous. Smoothness follows because in the smooth case all three are smooth (the zero section
by an earlier result, \(F\) by hypothesis, \(\pi'\) by the bundle structure).
Isomorphisms
A bijective bundle homomorphism whose inverse is also a bundle homomorphism is called a
bundle isomorphism. If such a map is also a diffeomorphism it is a
smooth bundle isomorphism, and the bundles are said to be
(smoothly) isomorphic. For smooth bundle homomorphisms covering the identity of a
common base, one need not verify smoothness of the inverse separately. Bijectivity plus smoothness
of \(F\) is enough, as the proposition below shows.
Definition: Bundle Isomorphism
A bundle isomorphism is a bijective bundle homomorphism whose inverse is also a
bundle homomorphism. A smooth bundle isomorphism is a bundle isomorphism that
is a diffeomorphism between smooth bundles. Two bundles \(E, E'\) are
(smoothly) isomorphic if there exists a (smooth) bundle isomorphism between
them.
Proposition: Bijective Smooth Bundle Homomorphisms Are Isomorphisms
Let \(E\) and \(E'\) be smooth vector bundles over a smooth manifold \(M\) with or without
boundary, and suppose \(F : E \to E'\) is a bijective smooth bundle homomorphism covering
\(\mathrm{Id}_M\). Then \(F\) is a smooth bundle isomorphism.
Proof:
It is enough to show that \(F^{-1}\) is smooth. Bijectivity and the bundle-homomorphism
property of \(F^{-1}\) (covering \(\mathrm{Id}_M\), fiberwise linear as the inverse of a linear
isomorphism) are immediate from the hypotheses on \(F\). Smoothness is a local property, so it
suffices to verify it on a neighborhood of each point of \(E'\). Write \(\pi\) and \(\pi'\) for
the projections of \(E\) and \(E'\). For each \(q \in M\), the restriction
\(F|_{E_q} : E_q \to E'_q\) is a linear bijection, so \(E\) and \(E'\) have a common rank
\(k\). Fix \(p \in M\). Choose smooth local trivializations
\(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) of \(E\) and
\(\Phi' : (\pi')^{-1}(U) \to U \times \mathbb{R}^k\) of \(E'\) on a common open neighborhood
\(U\) of \(p\) (such a common \(U\) exists because both bundles are trivializable in a
neighborhood of \(p\), so we may intersect the two trivializing sets).
The composite \(\Phi' \circ F \circ \Phi^{-1} : U \times \mathbb{R}^k \to U \times \mathbb{R}^k\)
is a smooth map (because \(\Phi^{-1}\), \(F\), and \(\Phi'\) are smooth) that covers
\(\mathrm{Id}_U\) and is fiberwise linear, so it has the form
\[
\Phi' \circ F \circ \Phi^{-1}(q, v) = \bigl(q,\, A(q) v\bigr)
\]
for a smooth map \(A : U \to GL(k, \mathbb{R})\). Smoothness of \(A\) follows from smoothness
of the composite by evaluating on standard basis vectors, as in the
transition function lemma.
The fiberwise restriction is invertible because \(F\) is bijective, so the values lie in
\(GL(k, \mathbb{R})\). Because matrix inversion is a smooth map
\(GL(k, \mathbb{R}) \to GL(k, \mathbb{R})\), the map \(q \mapsto A(q)^{-1}\) is smooth, and the
composite \(\Phi \circ F^{-1} \circ (\Phi')^{-1}(q, w) = (q, A(q)^{-1} w)\) is smooth on
\(U \times \mathbb{R}^k\). Hence \(F^{-1}\) is smooth on \((\pi')^{-1}(U)\), and since every
point of \(E'\) lies above some such \(U\), \(F^{-1}\) is smooth.
The over-\(M\) case
When both bundles have the same base \(M\), one usually restricts attention to bundle homomorphisms
covering the identity. These are the morphisms in the category of vector bundles over a fixed base.
Definition: Bundle Homomorphism over M
Let \(\pi : E \to M\) and \(\pi' : E' \to M\) be vector bundles over a common base \(M\). A
bundle homomorphism over \(M\) is a bundle homomorphism \(F : E \to E'\)
covering the identity \(\mathrm{Id}_M\). Equivalently, \(\pi' \circ F = \pi\), and the
restriction \(F|_{E_p} : E_p \to E'_p\) is linear for each \(p \in M\). When the bundles and
\(F\) are smooth, \(F\) is a smooth bundle homomorphism over \(M\).
Examples
Proposition: Smoothly Trivial Bundles and the Product Bundle
A smooth rank-\(k\) vector bundle over \(M\) is smoothly trivial if and only if it is smoothly
isomorphic over \(M\) to the product bundle \(M \times \mathbb{R}^k\).
Proof:
A smooth global trivialization \(\Phi : E \to M \times \mathbb{R}^k\) is, by definition of the
trivialization data, a smooth bundle homomorphism over \(M\) that is also a diffeomorphism,
hence a smooth bundle isomorphism. Indeed, \(\Phi\) satisfies \(\pi_M \circ \Phi = \pi\), where
\(\pi_M : M \times \mathbb{R}^k \to M\) is the projection onto the first factor, and it
restricts to a vector space isomorphism on each fiber. Conversely, a smooth bundle isomorphism
\(F : E \to M \times \mathbb{R}^k\) over \(M\) is a diffeomorphism satisfying the conditions of
a smooth global trivialization, so \(E\) is smoothly trivial.
Proposition: The Global Differential as a Bundle Homomorphism
Let \(F : M \to N\) be a smooth map between smooth manifolds with or without boundary. The
global differential \(dF : TM \to TN\), defined by
\(dF(v) = dF_p(v)\) for \(v \in T_pM\), is a smooth bundle homomorphism covering \(F\).
Proof:
Smoothness of \(dF\) is one of the
properties of the global differential.
Let \(\pi : TM \to M\) and \(\pi' : TN \to N\) be the tangent bundle projections. The base
relation \(\pi' \circ dF = F \circ \pi\) holds because \(dF\) sends \(T_pM\) to \(T_{F(p)}N\).
The restriction \(dF_p : T_pM \to T_{F(p)}N\) is a linear map by construction.
For a smooth vector bundle \(E \to M\) and an immersed or embedded submanifold \(S \subseteq M\),
the inclusion of the
restricted bundle
\(E|_S \hookrightarrow E\) is a smooth bundle homomorphism covering the inclusion
\(S \hookrightarrow M\). Its restriction to each fiber \(E_p\) (\(p \in S\)) is the identity, which
is linear, and smoothness follows because \(S \hookrightarrow M\) is smooth and the local
trivializations of \(E|_S\) are obtained by restricting those of \(E\).
Section Maps and the Characterization Lemma
A smooth bundle homomorphism over \(M\) induces an operation on sections. Composing a section of
the domain bundle with the homomorphism produces a section of the codomain bundle. The resulting
map between section spaces is not merely \(\mathbb{R}\)-linear but linear over \(C^\infty(M)\),
because multiplication by a base function commutes with the operation.
In fact, this stronger linearity property characterizes the section maps that come from smooth
bundle homomorphisms. Every \(C^\infty(M)\)-linear map between the
spaces of smooth sections
arises from a smooth bundle homomorphism over \(M\). Working with these homomorphisms and working
with \(C^\infty(M)\)-linear maps on section spaces are two views of the same object.
The induced section map
Let \(F : E \to E'\) be a smooth bundle homomorphism over a smooth manifold \(M\). For any smooth
section \(\sigma \in \Gamma(E)\), the composite
\[
\widetilde F(\sigma)(p) = F(\sigma(p)) , \quad p \in M ,
\]
defines a continuous map \(\widetilde F(\sigma) : M \to E'\) satisfying
\(\pi' \circ \widetilde F(\sigma)(p) = \pi'(F(\sigma(p))) = \pi(\sigma(p)) = p\), so it is a section
of \(E'\). Smoothness follows from smoothness of \(F\) and \(\sigma\), so
\(\widetilde F(\sigma) \in \Gamma(E')\). The assignment \(\sigma \mapsto \widetilde F(\sigma)\) is
the section map induced by \(F\).
The fiberwise linearity of \(F\) makes the section map \(\mathbb{R}\)-linear. Pointwise
multiplication by smooth functions on \(M\) also commutes with \(F\), because at each \(p\) it is
multiplication by the scalar \(u(p)\), which commutes with the linear map \(F|_{E_p}\). The
stronger property below follows.
Definition: Linearity over \(C^\infty(M)\)
A map \(\mathcal{F} : \Gamma(E) \to \Gamma(E')\) between spaces of smooth sections of vector
bundles over \(M\) is called linear over \(C^\infty(M)\) if for all
\(u_1, u_2 \in C^\infty(M)\) and \(\sigma_1, \sigma_2 \in \Gamma(E)\),
\[
\mathcal{F}(u_1 \sigma_1 + u_2 \sigma_2) = u_1 \mathcal{F}(\sigma_1) + u_2 \mathcal{F}(\sigma_2) .
\]
Linearity over \(C^\infty(M)\) is strictly stronger than \(\mathbb{R}\)-linearity. The latter
allows only constant scalars, whereas the former allows scalars that vary smoothly across the base. The
distinction is what cuts down general linear maps between section spaces to those induced by bundle
homomorphisms.
The characterization lemma
Lemma: Bundle Homomorphism Characterization
Let \(\pi : E \to M\) and \(\pi' : E' \to M\) be smooth vector bundles over a smooth manifold
\(M\) with or without boundary, and let
\(\mathcal{F} : \Gamma(E) \to \Gamma(E')\) be a map between their spaces of smooth sections.
Then \(\mathcal{F}\) is linear over \(C^\infty(M)\) if and only if there exists a smooth bundle
homomorphism \(F : E \to E'\) over \(M\) such that
\(\mathcal{F}(\sigma) = F \circ \sigma\) for all \(\sigma \in \Gamma(E)\). When such an \(F\)
exists, it is uniquely determined by \(\mathcal{F}\).
Proof:
The forward direction was noted above. If \(\mathcal{F} = \widetilde F\) is the section map
induced by a smooth bundle homomorphism \(F\) over \(M\), then \(\mathcal{F}\) is linear over
\(C^\infty(M)\). The substance of the lemma is the converse, which we prove now. Suppose
\(\mathcal{F} : \Gamma(E) \to \Gamma(E')\) is linear over \(C^\infty(M)\). We construct a smooth
bundle homomorphism \(F : E \to E'\) inducing it.
Step 1: \(\mathcal{F}\) acts locally.
We show that if
\(\rho_1, \rho_2 \in \Gamma(E)\) agree on some open set \(U \subseteq M\), then
\(\mathcal{F}(\rho_1)\) and \(\mathcal{F}(\rho_2)\) agree on \(U\). By linearity it suffices
to show that if \(\tau \in \Gamma(E)\) vanishes on \(U\), then \(\mathcal{F}(\tau)\) vanishes on
\(U\). Fix \(p \in U\) and choose a smooth bump function \(\psi \in C^\infty(M)\) supported in
\(U\) with \(\psi(p) = 1\). Then \(\psi \tau \equiv 0\) on \(M\) (because \(\tau\) vanishes on
the support of \(\psi\)), so by linearity
\[
0 = \mathcal{F}(\psi \tau) = \psi \, \mathcal{F}(\tau) ,
\]
and evaluating at \(p\) gives \(0 = \psi(p) \, \mathcal{F}(\tau)(p) = \mathcal{F}(\tau)(p)\).
Since this holds for every \(p \in U\), \(\mathcal{F}(\tau)\) vanishes on \(U\).
Step 2: \(\mathcal{F}\) acts pointwise.
We show that if
\(\rho_1, \rho_2 \in \Gamma(E)\) agree at a single point \(p\), then
\(\mathcal{F}(\rho_1)(p) = \mathcal{F}(\rho_2)(p)\). Again it suffices to show that if
\(\tau \in \Gamma(E)\) satisfies \(\tau(p) = 0\), then \(\mathcal{F}(\tau)(p) = 0\). Let \(k\) be
the rank of \(E\) and choose a
smooth local frame \((\sigma_1, \dots, \sigma_k)\) for \(E\) on some neighborhood
\(W\) of \(p\), and write \(\tau = u^i \sigma_i\) on \(W\) for some smooth functions
\(u^i \in C^\infty(W)\). The condition \(\tau(p) = 0\) gives
\(u^1(p) = \dots = u^k(p) = 0\). By the
extension lemma for vector bundles
applied to each \(\sigma_i\) (and the
extension lemma for smooth functions
applied to each \(u^i\)), there exist global smooth sections
\(\widetilde \sigma_i \in \Gamma(E)\) and global smooth functions
\(\widetilde u^i \in C^\infty(M)\) that agree respectively with \(\sigma_i\) and \(u^i\) on some
possibly smaller neighborhood of \(p\). Shrinking that neighborhood if necessary, we may
assume \(\tau = \widetilde u^i \widetilde \sigma_i\) holds on it. Then by Step 1 and
linearity over \(C^\infty(M)\),
\[
\mathcal{F}(\tau)(p) = \mathcal{F}(\widetilde u^i \widetilde \sigma_i)(p)
= \widetilde u^i(p) \, \mathcal{F}(\widetilde \sigma_i)(p)
= u^i(p) \, \mathcal{F}(\widetilde \sigma_i)(p) = 0 .
\]
Step 3: defining \(F\).
Steps 1 and 2 show that the value \(\mathcal{F}(\sigma)(p)\)
depends only on the value \(\sigma(p) \in E_p\), not on the global section \(\sigma\) chosen to
produce it. By
every element of \(E\) is on a smooth global section,
for each \(v \in E\) with \(p = \pi(v)\) there exists \(\widetilde v \in \Gamma(E)\) with
\(\widetilde v(p) = v\). Define
\[
F(v) = \mathcal{F}(\widetilde v)(p) \in E'_p .
\]
Step 2 shows this is independent of the choice of \(\widetilde v\). The relation
\(\pi' \circ F = \pi\) is built in (\(F(v) \in E'_p\) where \(p = \pi(v)\)), and fiberwise
linearity follows from the \(\mathbb{R}\)-linearity of \(\mathcal{F}\). For
\(v, w \in E_p\) and \(a, b \in \mathbb{R}\), choose \(\widetilde v, \widetilde w \in \Gamma(E)\)
with the prescribed values at \(p\). Then \(a \widetilde v + b \widetilde w \in \Gamma(E)\) has
value \(av + bw\) at \(p\), and
\(F(av + bw) = \mathcal{F}(a \widetilde v + b \widetilde w)(p) = a \mathcal{F}(\widetilde v)(p) + b \mathcal{F}(\widetilde w)(p) = aF(v) + bF(w)\).
The identity \(\mathcal{F}(\sigma) = F \circ \sigma\) holds by construction.
Step 4: smoothness of \(F\).
Smoothness is a local property. We show \(F\) is
smooth on a neighborhood of each point \(v_0 \in E\). Let \(p = \pi(v_0)\), and choose a smooth
local frame \((\sigma_i)\) for \(E\) on a neighborhood \(W\) of \(p\) and a smooth local frame
\((\sigma'_j)\) for \(E'\) on a (possibly smaller) common neighborhood. Using the extension
lemma, replace \((\sigma_i)\) by global smooth sections \((\widetilde \sigma_i)\) agreeing with
\((\sigma_i)\) on some neighborhood \(U \subseteq W\) of \(p\), and analogously for the
\(\sigma'_j\). Let \(m\) be the rank of \(E'\). On \(U\) there exist smooth
functions \(A_i^j \in C^\infty(U)\) such that
\(\mathcal{F}(\widetilde \sigma_i)|_U = A_i^j \sigma'_j\), since the \((\sigma'_j)\) form a
frame for \(E'\) on \(U\) and any smooth section there is uniquely expressible in this form by
the
local frame criterion for smoothness.
For any \(q \in U\) and \(v = v^i \sigma_i(q) \in E_q\), the global section
\(v^i \widetilde \sigma_i \in \Gamma(E)\) has value \(v\) at \(q\), so by construction
\[
F(v) = \mathcal{F}(v^i \widetilde \sigma_i)(q) = v^i \mathcal{F}(\widetilde \sigma_i)(q)
= v^i A_i^j(q) \sigma'_j(q) \in E'_q .
\]
Let \(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) and
\(\Phi' : (\pi')^{-1}(U) \to U \times \mathbb{R}^m\) be the smooth local trivializations
associated with the frames \((\sigma_i)\) and \((\sigma'_j)\). With respect to these
trivializations,
\[
\Phi' \circ F \circ \Phi^{-1}\bigl(q, (v^1, \dots, v^k)\bigr)
= \bigl(q,\, (A_i^1(q) v^i, \dots, A_i^m(q) v^i)\bigr) ,
\]
which is smooth in \((q, v)\) because the \(A_i^j\) are smooth in \(q\). Since
\(\Phi, \Phi'\) are diffeomorphisms, \(F\) is smooth on \(\pi^{-1}(U)\). This holds in a
neighborhood of every point, so \(F\) is smooth on \(E\).
Uniqueness.
If \(F_1, F_2 : E \to E'\) are smooth bundle homomorphisms over
\(M\) with \(F_1 \circ \sigma = F_2 \circ \sigma\) for every \(\sigma \in \Gamma(E)\), then for
each \(v \in E\) with \(p = \pi(v)\), choosing \(\sigma \in \Gamma(E)\) with \(\sigma(p) = v\)
gives \(F_1(v) = F_1(\sigma(p)) = F_2(\sigma(p)) = F_2(v)\). Hence \(F_1 = F_2\).
Why \(C^\infty(M)\)-linearity matters
The characterization lemma is the algebraic backbone of the entire theory of tensor fields and
related "tensorial" geometric structures, which later pages of the manifold series develop. A
Riemannian metric on \(M\) can be presented as a symmetric \(C^\infty(M)\)-bilinear map
\(g : \Gamma(TM) \times \Gamma(TM) \to C^\infty(M)\). We identify \(C^\infty(M)\) with the
smooth sections of the trivial line bundle \(M \times \mathbb{R}\). Running the proof of the
lemma in each slot of \(g\) then shows that such a map comes from a bilinear form on each
\(T_pM\). Positive-definiteness becomes a condition imposed fiber by fiber. The same passage
from sections to fibers applies to tensor fields defined via section calculus.
Among \(\mathbb{R}\)-linear maps between section spaces, linearity over \(C^\infty(M)\) is
exactly the algebraic condition that distinguishes "tensorial" objects from operators whose
value at a point depends on more than the value of a section at that point. Differentiation is
an operator of the second kind. It is \(\mathbb{R}\)-linear but not \(C^\infty(M)\)-linear
because of the Leibniz rule. Connections (not defined on this page) fall on the non-tensorial
side of this divide, because they satisfy a Leibniz rule. The lemma thus separates the
tensorial from the differential, and is used implicitly whenever a geometric structure is
defined by a multilinear map on section spaces.
The characterization also licenses an abuse of notation. The bundle homomorphism \(F\) and its induced
section map \(\widetilde F\) are often denoted by the same letter \(F\), with context disambiguating
pointwise application from action on sections. The
scalar multiplication
\(X \mapsto aX\) on vector fields, for example, comes from the bundle homomorphism \(TM \to TM\)
that multiplies each tangent vector by the constant \(a\), and the same symbol is used for both.
As noted above, differential operators that involve derivation are not bundle homomorphisms,
because they are \(\mathbb{R}\)-linear but not \(C^\infty(M)\)-linear (the Leibniz rule introduces
a derivative of \(u\) when one multiplies by \(u \in C^\infty(M)\)). The most prominent example, on
a manifold without boundary where it is defined, is the
Lie derivative
\(\mathcal{L}_X : \mathfrak{X}(M) \to \mathfrak{X}(M)\) along a vector field \(X\) that is not
identically zero. Here \(\mathfrak{X}(M) = \Gamma(TM)\) is the space of smooth vector fields.
Subbundles
A subbundle is to a vector bundle what a subspace is to a vector space: a sub-collection of the
fibers that fits together coherently. The definition packages two conditions. The subbundle must be
a vector bundle in its own right, and it must sit inside the ambient bundle as a topological
subspace whose fibers are linear subspaces. A smooth bundle homomorphism over \(M\) of constant
rank, a notion made precise in the next section, produces smooth subbundles as its kernel and
image. The local-frame criterion proved here produces subbundles from local-section data.
Definition: Subbundle
Given a vector bundle \(\pi_E : E \to M\), a subbundle of \(E\) is a vector
bundle \(\pi_D : D \to M\) in which \(D\) is a topological subspace of \(E\) and
\(\pi_D = \pi_E|_D\) is the restriction of \(\pi_E\) to \(D\), such that for each \(p \in M\)
the subset \(D_p = D \cap E_p\) is a linear subspace of \(E_p\), with the vector space structure
on \(D_p\) inherited from \(E_p\). The condition that \(D\) be a vector bundle over \(M\) forces
all fibers \(D_p\) to be nonempty and of the same dimension.
Definition: Smooth Subbundle
If \(\pi_E : E \to M\) is a smooth vector bundle, a subbundle \(D \subseteq E\) is called a
smooth subbundle if it is itself a smooth vector bundle and \(D\) is an
embedded submanifold (with or without boundary) of \(E\).
Proposition: The Inclusion of a Smooth Subbundle Is a Bundle Homomorphism
Let \(\pi : E \to M\) be a smooth vector bundle and let \(D \subseteq E\) be a smooth subbundle.
The inclusion map \(\iota : D \hookrightarrow E\) is a smooth bundle homomorphism over \(M\).
Proof:
The smoothness of \(\iota\) follows from \(D\) being an embedded submanifold (with or without
boundary) of \(E\) (the inclusion of such a submanifold into the ambient manifold is smooth by
definition). The relation \(\pi \circ \iota = \pi_D\) (where \(\pi_D = \pi|_D\) is the
projection of \(D\)) holds pointwise, exhibiting \(\iota\) as a map covering the identity
\(\mathrm{Id}_M\). Fiberwise, \(\iota|_{D_p} : D_p \hookrightarrow E_p\) is the inclusion of a
linear subspace, which is linear.
The local-frame criterion
Constructing a subbundle by hand requires verifying that a candidate union
\(D = \bigcup_p D_p\) of linear subspaces is itself a vector bundle in the right sense. A topology
must be built, charts must be checked, and the bundle structure must match the ambient one. The
next criterion reduces this entire verification to producing locally a smooth basis. If smooth local
sections spanning each \(D_p\) exist on a neighborhood of every base point, the union is
automatically a smooth subbundle.
Lemma: Local Frame Criterion for Subbundles
Let \(\pi : E \to M\) be a smooth vector bundle, and suppose that for each \(p \in M\) we are
given an \(m\)-dimensional linear subspace \(D_p \subseteq E_p\). Then
\(D = \bigcup_{p \in M} D_p \subseteq E\) is a smooth subbundle of \(E\) of rank \(m\) if and
only if the following condition holds:
Each point of \(M\) has a neighborhood \(U\) on which there exist smooth local sections
\(\sigma_1, \dots, \sigma_m : U \to E\) of the ambient bundle such that
\((\sigma_1(q), \dots, \sigma_m(q))\) is a basis of \(D_q\) for every \(q \in U\).
When the condition holds, \(D\) carries a smooth subbundle structure in which, whenever
\(U \subseteq M\) is open and \(\sigma_1, \dots, \sigma_m\) are smooth local sections of \(E\)
over \(U\) whose values at each \(q \in U\) form a basis of \(D_q\), the family
\((\sigma_1, \dots, \sigma_m)\) is a smooth local frame for \(D\) over \(U\).
Proof:
Necessity.
Suppose \(D\) is a smooth subbundle of \(E\). For each \(p \in M\),
\(D\) admits a smooth local trivialization over some neighborhood \(U\) of \(p\), and the
frame associated with this trivialization
produces smooth local sections \(\tau_1, \dots, \tau_m : U \to D\) whose values form a basis of
\(D_q\) at each \(q \in U\). Composing with the inclusion \(\iota : D \hookrightarrow E\) (a
smooth bundle homomorphism by the proposition above), the sections
\(\sigma_i = \iota \circ \tau_i : U \to E\) are smooth local sections of \(E\) with the same
values, and their values continue to form a basis of \(D_q\) at each \(q \in U\). The
local-section condition is satisfied.
Sufficiency.
Suppose the local-section condition holds. We construct a
smooth vector bundle structure on \(D\) and verify that \(D\) is an embedded submanifold (with
or without boundary) of \(E\), which together make \(D\) a smooth subbundle. Let \(E\) have
rank \(k \ge m\).
Fix \(p \in M\) and let \(U \ni p\) be a neighborhood on which smooth local sections
\(\sigma_1, \dots, \sigma_m\) of \(E\) form a basis of \(D_q\) for each \(q \in U\).
When \(1 \le m \lt k\), the
completion of local frames
(applied to \(E\) and the partial frame \((\sigma_1, \dots, \sigma_m)\), shrinking \(U\) if
necessary) gives smooth sections \(\sigma_{m+1}, \dots, \sigma_k\) on \(U\) such that
\((\sigma_1, \dots, \sigma_k)\) is a smooth local frame for \(E\) over \(U\). When \(m = k\),
the sections \(\sigma_1, \dots, \sigma_k\) already form such a frame. When \(m = 0\), the
family \((\sigma_1, \dots, \sigma_m)\) is empty and \(D_q = \{0\}\). After shrinking \(U\), we
take for \((\sigma_1, \dots, \sigma_k)\) the frame associated with a smooth local
trivialization of \(E\) over \(U\). Let \(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) be the
smooth local trivialization of \(E\) associated with the frame \((\sigma_1, \dots, \sigma_k)\),
as given by the
frame \(\leftrightarrow\) trivialization correspondence:
\[
\Phi\!\left(s^1 \sigma_1(q) + \dots + s^k \sigma_k(q)\right) = \bigl(q,\, (s^1, \dots, s^k)\bigr) .
\]
Under \(\Phi\), the set \(D \cap \pi^{-1}(U)\) is sent to
\(\{(q, (s^1, \dots, s^m, 0, \dots, 0)) : q \in U,\ s^i \in \mathbb{R}\}\), because a vector
\(s^1\sigma_1(q) + \dots + s^k\sigma_k(q) \in E_q\) lies in \(D_q\) if and only if it lies in
the span of \((\sigma_1(q), \dots, \sigma_m(q))\), which is to say
\(s^{m+1} = \dots = s^k = 0\). The image \(U \times \mathbb{R}^m \times \{0\}^{k-m}\) is the
image of the smooth embedding \((q, s) \mapsto (q, (s, 0))\) of \(U \times \mathbb{R}^m\) into
\(U \times \mathbb{R}^k\), hence an embedded submanifold (with or without boundary) of
\(U \times \mathbb{R}^k\), by the result that
images of embeddings are submanifolds
when \(U\) contains no boundary point of \(M\), and by the result on
images of embeddings with boundary
otherwise. Therefore \(D \cap \pi^{-1}(U)\) is an embedded submanifold (with or without
boundary) of \(\pi^{-1}(U) \subseteq E\). Its smooth structure is the one for which
\((q, s) \mapsto \Phi^{-1}(q, (s, 0))\) is a diffeomorphism from \(U \times \mathbb{R}^m\) onto
it. The sets \(D \cap \pi^{-1}(U)\) obtained in this way from all \(p \in M\) are open in
\(D\), because each \(\pi^{-1}(U)\) is open in \(E\), and they cover \(D\). Their smooth
structures agree on overlaps. Let \(U'\) and \(\Phi'\) arise in the same way from a second
choice of sections, with \(U \cap U' \neq \emptyset\). The
transition lemma
gives \(\Phi' \circ \Phi^{-1}(q, v) = (q, \rho(q) v)\) for
\((q, v) \in (U \cap U') \times \mathbb{R}^k\), with \(\rho : U \cap U' \to GL(k, \mathbb{R})\)
smooth. Both \(\Phi\) and \(\Phi'\) send \(D \cap \pi^{-1}(U \cap U')\) onto the set where the
last \(k - m\) fiber coordinates vanish, so \(\rho(q)(s, 0) = (s', 0)\) for some
\(s' \in \mathbb{R}^m\). The change between the two parametrizations of
\(D \cap \pi^{-1}(U \cap U')\) is therefore \((q, s) \mapsto (q, s')\), which is smooth because
\(s'\) consists of the first \(m\) entries of \(\rho(q)(s, 0)\). Exchanging the roles of
\(\Phi\) and \(\Phi'\) shows that the inverse of this change of parametrization is smooth as
well, so the two structures coincide on the overlap. Since these open pieces cover \(D\) and
their structures agree on overlaps, their smooth charts together form a smooth atlas on \(D\).
With the subspace topology, \(D\) is Hausdorff and second countable because \(E\) is, so this
atlas makes \(D\) a smooth manifold (with or without boundary). The inclusion
\(D \hookrightarrow E\) is a smooth immersion, because on each open piece
\(D \cap \pi^{-1}(U)\) it restricts to a smooth embedding, and it is a topological embedding
because \(D\) carries the subspace topology. It is therefore a smooth embedding. Hence \(D\) is
an embedded submanifold (with or without boundary) of \(E\).
The map \(\Psi : D \cap \pi^{-1}(U) \to U \times \mathbb{R}^m\) defined by
\[
\Psi\!\left(s^1 \sigma_1(q) + \dots + s^m \sigma_m(q)\right) = \bigl(q,\, (s^1, \dots, s^m)\bigr)
\]
is the composition of two maps. The first is the restriction of \(\Phi\) to
\(D \cap \pi^{-1}(U)\). It sends \(\Phi^{-1}(q, (s, 0))\) to \((q, (s, 0))\) and is therefore,
for the smooth structure fixed above, a diffeomorphism onto
\(U \times \mathbb{R}^m \times \{0\}^{k-m}\). The second is the map
\(U \times \mathbb{R}^m \times \{0\}^{k-m} \to U \times \mathbb{R}^m\) that drops the trailing
zeros, which is also a diffeomorphism. Hence \(\Psi\) is a diffeomorphism, fiberwise linear,
and satisfies
\
\pi_U \circ \Psi = \pi|_D.
\]
Therefore \(\Psi\) is a smooth local trivialization of \(D\) over \(U\). Its associated frame is
\((\sigma_1, \dots, \sigma_m)\), because \(\Psi^{-1}(q, e_i) = \sigma_i(q)\) for the standard basis vectors \(e_i\) of
\(\mathbb{R}^m\), so the sections \(\sigma_i\) form a smooth local frame for \(D\) over \(U\)
(the empty frame when \(m = 0\)). The construction can be started at any point of an open set
on which sections with this basis property are given, using those same sections, so near each
point of that set they are smooth as maps into \(D\). Since smoothness is local and the basis
property holds at each point, they form a smooth local frame for \(D\) over the whole set.
Every point of \(M\) lies in a neighborhood over which such a trivialization \(\Psi\) is
constructed, so \(D\) inherits a smooth vector bundle structure of rank \(m\) over \(M\), and
the inclusion \(D \hookrightarrow E\) is the inclusion of an embedded submanifold (with or
without boundary). Thus \(D\) is a smooth subbundle of \(E\).
Examples of subbundles
(a) The span of a nowhere-vanishing vector field.
Let \(M\) be a smooth
manifold and \(V\) a nowhere-vanishing smooth vector field on \(M\). The set \(D \subseteq TM\)
whose fiber at each \(p \in M\) is the line \(\mathrm{span}(V_p) \subseteq T_pM\) is a smooth
rank-1 subbundle of \(TM\). The local-section condition is satisfied with the single section
\(\sigma_1 = V\) on \(U = M\).
(b) Span of part of a global frame.
Suppose \(E \to M\) is a trivial smooth
vector bundle with smooth global frame \((\sigma_1, \dots, \sigma_k)\). For \(0 \le m \le k\),
the union \(D \subseteq E\) with fiber
\(D_p = \mathrm{span}(\sigma_1(p), \dots, \sigma_m(p))\) is a smooth rank-\(m\)
subbundle of \(E\). The local-section condition holds globally with the first \(m\)
members of the frame. When \(M = G\) is a Lie group and
\((\sigma_1, \dots, \sigma_n)\) is the
basis of left-invariant vector fields
corresponding to a basis of \(\mathrm{Lie}(G)\), every \(m\)-dimensional subspace of
\(\mathrm{Lie}(G)\) determines a smooth rank-\(m\) left-invariant subbundle of \(TG\). These
are the fundamental examples of "distributions" in the integrability theory of vector fields.
(c) The tangent bundle of a submanifold.
Let \(M\) be a smooth manifold
with or without boundary and \(S \subseteq M\) an immersed \(k\)-submanifold, with or without
boundary. The
tangent bundle
\(TS\) sits inside the
ambient tangent bundle
\(TM|_S\) as a smooth rank-\(k\) subbundle. To see this, fix \(q_0 \in S\) and choose a smooth
chart \((V, \varphi)\) for \(S\) around \(q_0\), with coordinate functions
\((x^1, \dots, x^k)\). Write \(\iota : S \hookrightarrow M\) for the inclusion. The
images \(d\iota\bigl(\partial/\partial x^i|_q\bigr)\) of the coordinate vector fields
are smooth local sections of \(TM|_S\) over \(V\), because \(\iota\) is smooth and its
global differential is a smooth bundle homomorphism, and their values span
\(T_qS\), regarded as a
subspace of \(T_qM\),
at each \(q \in V\). The local-section condition holds, and the
criterion gives \(TS\) the structure of a smooth subbundle.
Kernel and Image Subbundles
The most productive source of subbundles is the kernel-and-image construction for bundle
homomorphisms. For a linear map between finite-dimensional vector spaces the kernel and image are
immediately subspaces. For a bundle homomorphism, taking kernel and image fiberwise gives
candidate subbundles of the domain and codomain. The catch is that the dimension of the kernel and
the rank of the image are not, in general, constant along the base, and can jump from point to
point. When they do not jump, the candidates are genuine smooth subbundles. When they do, the
analysis breaks down. The relevant notion is constant rank.
The rank of a bundle homomorphism
Definition: Rank of a Bundle Homomorphism
Let \(F : E \to E'\) be a bundle homomorphism over \(M\). For each \(p \in M\), the
rank of \(F\) at \(p\) is the rank of the linear map
\(F|_{E_p} : E_p \to E'_p\). The map \(F\) is said to have constant rank if
the rank of \(F\) at \(p\) is the same for all \(p \in M\). This common value is the
rank of \(F\).
The constant-rank condition is a natural strengthening of the bundle-homomorphism axioms. It ensures
that the fiberwise kernel and image structures fit together coherently across the base. Without it,
fiber dimensions vary and no bundle structure can survive.
The constant-rank theorem
Theorem: Kernel and Image as Subbundles Under Constant Rank
Let \(E\) and \(E'\) be smooth vector bundles over a smooth manifold \(M\) with or
without boundary, and let \(F : E \to E'\) be a smooth bundle homomorphism over \(M\). Define
\[
\begin{align*}
\mathrm{Ker}\, F &= \bigcup_{p \in M} \mathrm{Ker}(F|_{E_p}) \subseteq E, \\\\
\mathrm{Im}\, F &= \bigcup_{p \in M} \mathrm{Im}(F|_{E_p}) \subseteq E'.
\end{align*}
\]
Then \(\mathrm{Ker}\, F\) and \(\mathrm{Im}\, F\) are smooth subbundles of \(E\) and \(E'\)
respectively if and only if \(F\) has constant rank.
Proof:
Necessity.
If \(\mathrm{Ker}\, F\) and \(\mathrm{Im}\, F\) are smooth
subbundles, their fibers have constant dimension by the definition of a vector bundle. The fiber
\((\mathrm{Im}\, F)_p\) has dimension equal to the rank of \(F\) at \(p\), and the fiber
\((\mathrm{Ker}\, F)_p\) has dimension \(\dim E_p - \mathrm{rank}(F|_{E_p})\). Constancy of
either dimension along \(M\) forces the rank of \(F\) to be constant.
Sufficiency.
Suppose \(F\) has constant rank \(r\), and let \(k\) denote
the rank of \(E\), so the fibers of \(\mathrm{Ker}\, F\) have dimension \(k - r\) and those of
\(\mathrm{Im}\, F\) have dimension \(r\). We show that each of \(\mathrm{Im}\, F\) and
\(\mathrm{Ker}\, F\) satisfies the local-frame criterion for subbundles.
Image.
Fix \(p \in M\), and choose a smooth local frame
\((\sigma_1, \dots, \sigma_k)\) for \(E\) over a neighborhood \(U\) of \(p\). The sections
\(F \circ \sigma_1, \dots, F \circ \sigma_k\) of \(E'\) over \(U\) span \(\mathrm{Im}(F|_{E_q})\)
at each \(q \in U\), and some \(r\) of them are linearly independent at \(p\) (the rank of
\(F|_{E_p}\) being \(r\)). By relabeling the indices if necessary, we may assume
\((F \circ \sigma_1, \dots, F \circ \sigma_r)\) are linearly independent at \(p\). Linear
independence is an open condition, so there is a possibly smaller neighborhood
\(U_0 \subseteq U\) of \(p\) on which \((F \circ \sigma_1, \dots, F \circ \sigma_r)\) remain
linearly independent at every point. By the constant-rank hypothesis, the image
\(\mathrm{Im}(F|_{E_q}) = (\mathrm{Im}\, F)_q\) has dimension exactly \(r\) for every
\(q \in U_0\), so \((F \circ \sigma_1, \dots, F \circ \sigma_r)\) are not only linearly
independent but span \((\mathrm{Im}\, F)_q\) at each point of \(U_0\). The local-frame criterion
for subbundles applies, and \(\mathrm{Im}\, F\) is a smooth rank-\(r\) subbundle of \(E'\).
Kernel.
Let \(U_0\) and \((\sigma_1, \dots, \sigma_r)\) be as above. Taken as a
local frame on \(U_0\), these sections span a smooth rank-\(r\) subbundle
\(V \subseteq E|_{U_0}\) by the local-frame criterion. This subbundle is complementary to
\(\mathrm{Ker}\, F\) in the following sense. The restriction
\(F|_V : V \to (\mathrm{Im}\, F)|_{U_0}\) is bijective on each fiber, because
\((F \circ \sigma_i)\) is a basis of \((\mathrm{Im}\, F)_q\) and \((\sigma_i)\) is a basis
of \(V_q\), and it is a smooth bundle
homomorphism over \(U_0\). By the
bijective-bundle-homomorphism proposition,
\(F|_V\) is a smooth bundle isomorphism. Let \((F|_V)^{-1} : (\mathrm{Im}\, F)|_{U_0} \to V\)
be its inverse, and define
\[
\Psi : E|_{U_0} \to E|_{U_0}, \quad
\Psi(v) = v - (F|_V)^{-1}\bigl(F(v)\bigr) .
\]
Each term is a smooth bundle homomorphism over \(U_0\) (composition with the homomorphism
\(F\) from \(E|_{U_0}\) to \((\mathrm{Im}\, F)|_{U_0}\), then with \((F|_V)^{-1}\), each preserves
fibers and is linear), so \(\Psi\) is itself a smooth bundle homomorphism over \(U_0\).
We verify that \(\Psi\) takes its values in \((\mathrm{Ker}\, F)|_{U_0}\) and restricts to the
identity on \((\mathrm{Ker}\, F)|_{U_0}\). For any \(v \in E|_{U_0}\),
\[
F(\Psi(v)) = F(v) - F\bigl((F|_V)^{-1}(F(v))\bigr) = F(v) - F(v) = 0 ,
\]
since \((F|_V)^{-1}(F(v)) \in V\) and \(F|_V\) is the inverse of \((F|_V)^{-1}\) on \(V\). Hence
\(\Psi(v) \in (\mathrm{Ker}\, F)_{q}\) for \(q = \pi(v)\). Conversely, if
\(v \in (\mathrm{Ker}\, F)_q\), then \(F(v) = 0\), so
\(\Psi(v) = v - (F|_V)^{-1}(0) = v\). Thus \(\Psi\) is the identity on \((\mathrm{Ker}\, F)|_{U_0}\).
These two properties together say that \(\Psi\) is a fiberwise projection of \(E|_{U_0}\) onto
\(\mathrm{Ker}\, F\) along \(V\) (idempotent, \(\Psi \circ \Psi = \Psi\)).
The image of \(\Psi\) is therefore exactly \((\mathrm{Ker}\, F)|_{U_0}\). Since
\(V \oplus \mathrm{Ker}\, F\) recovers \(E|_{U_0}\) fiberwise (the rank-nullity theorem gives
\(\dim V_q + \dim (\mathrm{Ker}\, F)_q = r + (k - r) = k = \dim E_q\)), and the splitting is
smooth because \((F|_V)^{-1}\) is smooth, \((\mathrm{Ker}\, F)|_{U_0}\) is the image of a
smooth bundle homomorphism with constant rank \(k - r\). Applying the image-subbundle argument
already proved to \(\Psi\) shows \((\mathrm{Ker}\, F)|_{U_0}\) is a smooth rank-\((k-r)\)
subbundle of \(E|_{U_0}\). The same argument can be carried out on a neighborhood of every
point of \(M\), so \(\mathrm{Ker}\, F\) is a smooth rank-\((k-r)\) subbundle of \(E\).
The orthogonal complement and the normal bundle
The constant-rank theorem produces subbundles abstractly. For the special case of subbundles of
\(T\mathbb{R}^n\) restricted to a submanifold, the Euclidean inner product gives an explicit
geometric construction of a complementary subbundle: the orthogonal complement, fiber by fiber.
Lemma: Orthogonal Complement Bundles
Let \(M\) be an
immersed submanifold with or without boundary in \(\mathbb{R}^n\),
and let \(D\) be a smooth rank-\(k\) subbundle of the
ambient tangent bundle
\(T\mathbb{R}^n|_M\). For each \(p \in M\), let \(D_p^\perp\) denote the orthogonal complement
of \(D_p\) in \(T_p\mathbb{R}^n\) with respect to the Euclidean dot product, and let
\[
D^\perp = \{(p, v) \in T\mathbb{R}^n : p \in M,\ v \in D_p^\perp\} \subseteq T\mathbb{R}^n|_M .
\]
Then \(D^\perp\) is a smooth rank-\((n - k)\) subbundle of \(T\mathbb{R}^n|_M\). For each
\(p \in M\), there is a smooth orthonormal frame for \(D^\perp\) on a neighborhood of \(p\).
Proof:
Fix \(p \in M\), and let \((X_1, \dots, X_k)\) be a smooth local frame for \(D\) over some
neighborhood \(V\) of \(p\) in \(M\). Because immersed submanifolds are locally embedded,
without boundary
or
with boundary,
the
local slice criterion
or its
version with boundary
lets us shrink \(V\) so that it is a single slice or half-slice inside some open coordinate
ball \(U \subseteq \mathbb{R}^n\). A slice or half-slice is cut out of \(U\) by equations and
(for a half-slice) one non-strict inequality in the coordinates, so it is closed in \(U\).
Either criterion also lets us choose the chart so that the coordinates parametrizing the slice
restrict to coordinates of \(V\). Each \(X_i\) is then a smooth function of those coordinates
on the slice. In the half-slice case, it extends across the bounding hyperplane by the
half-space reading of smoothness. Reading \(X_i\) as a function of all \(n\) coordinates of
\(U\) gives the smooth local extension that the extension lemma requires. By the
extension lemma,
each \(X_i\) extends to a smooth section \(\widetilde X_i\) of \(T\mathbb{R}^n\) over
\(U\) agreeing with \(X_i\) on \(V\). These stay linearly independent near \(p\). Whenever
\(U\) is shrunk below, \(V\) is replaced by \(V \cap U\). When \(1 \le k \lt n\), after
shrinking \(U\), the
completion of local frames
supplies \(\widetilde X_{k+1}, \dots, \widetilde X_n\) making
\((\widetilde X_1, \dots, \widetilde X_n)\) a smooth local frame for \(T\mathbb{R}^n\) over
\(U\). When \(k = n\), the \(\widetilde X_i\) already form such a frame after shrinking \(U\).
When \(k = 0\), the coordinate vector fields of \(\mathbb{R}^n\) serve as
\(\widetilde X_1, \dots, \widetilde X_n\). Applying the
Gram-Schmidt process for frames
to \((\widetilde X_1, \dots, \widetilde X_n)\), we obtain a smooth orthonormal frame
\((E_1, \dots, E_n)\) for \(T\mathbb{R}^n\) over \(U\) such that
\(\mathrm{span}(E_1|_q, \dots, E_k|_q) = \mathrm{span}(X_1|_q, \dots, X_k|_q) = D_q\) for every
\(q \in V\) (the Gram-Schmidt process preserves the flag of nested spans).
The sections \((E_{k+1}, \dots, E_n)\) restricted to \(V\) form a smooth orthonormal frame for
\(D^\perp\). At each \(q \in V\), the vectors \(E_{k+1}|_q, \dots, E_n|_q\) are orthonormal and
orthogonal to every \(E_i|_q\) (\(i \le k\)), hence to all of \(D_q\), so they lie in
\(D_q^\perp\). They also span an \((n-k)\)-dimensional subspace, which by dimension count is all
of \(D_q^\perp\). The local-frame criterion for subbundles applies, and \(D^\perp\) is a smooth
rank-\((n - k)\) subbundle of \(T\mathbb{R}^n|_M\).
The most important special case is the
normal bundle
of a submanifold, defined earlier for an embedded submanifold of \(\mathbb{R}^n\) without
boundary as the orthogonal complement of \(TM\) inside \(T\mathbb{R}^n|_M\). The definition is
pointwise, \(N_pM = (T_pM)^\perp\) with \(T_pM\) regarded as a subspace of \(T_p\mathbb{R}^n\).
This subspace has the same dimension as \(M\) because the differential of the inclusion is
injective, as in the identification of
tangent spaces of submanifolds.
The definition therefore applies verbatim to an immersed submanifold with or without boundary, and
we use it in that generality below. Within the present framework, the normal bundle is the
orthogonal complement of a particular smooth subbundle.
Corollary: The Normal Bundle as a Subbundle
Let \(M \subseteq \mathbb{R}^n\) be an immersed \(m\)-dimensional
submanifold with or without boundary. The
normal bundle
\(NM\) is a smooth rank-\((n - m)\) subbundle of \(T\mathbb{R}^n|_M\), and each point of \(M\)
has a neighborhood over which there exists a smooth orthonormal frame for \(NM\).
Proof:
The tangent bundle \(TM\) is a smooth rank-\(m\) subbundle of \(T\mathbb{R}^n|_M\) by Example
(c) above. Applying the orthogonal-complement lemma to \(D = TM\) (which has rank \(m\)) gives
\(D^\perp = NM\) as a smooth rank-\((n - m)\) subbundle of \(T\mathbb{R}^n|_M\), with smooth
orthonormal local frames on neighborhoods of every point.
What constant rank buys
The constant-rank theorem produces subbundles algebraically rather than geometrically. Given a
bundle homomorphism with the right algebraic property (rank constant), the fiberwise kernel and
image sets automatically inherit smooth subbundle structure, without any further
geometric construction. This is in marked contrast to the local-frame criterion, which requires
explicit production of smooth sections spanning the candidate subbundle. The two routes are
complementary. When one has a bundle homomorphism in hand, the
kernel-and-image route applies. When one has only candidate fibers and needs to manufacture
local frames, the local-frame route applies. The orthogonal-complement specialization above
gives the normal bundle as a tangible geometric object. It is the same normal bundle that
supplied the tubular neighborhood theorem, recognized now as one instance of a general
construction inside the broader subbundle framework.
Fiber Bundles
Vector bundles are the most useful family of locally trivial fibrations, but they are not the only
one. Dropping the requirement that the fibers be vector spaces, and replacing it with the
requirement that the fibers be homeomorphic to a fixed model space, gives the broader notion of a
fiber bundle. Many natural geometric and topological constructions fall
outside the vector-bundle framework and yet fit naturally into this larger picture. Covering
spaces, principal bundles, frame bundles, associated bundles, and Hopf fibrations are among them,
and several recur in later pages. We introduce only the definitions and the most immediate
examples here.
Definition: Fiber Bundle
Let \(M\) and \(F\) be topological spaces. A fiber bundle over \(M\) with model fiber
\(F\) is a topological space \(E\) together with a surjective continuous map
\(\pi : E \to M\) such that for each \(x \in M\) there exist a neighborhood \(U\) of \(x\) in
\(M\) and a homeomorphism \(\Phi : \pi^{-1}(U) \to U \times F\) satisfying
\(\pi_1 \circ \Phi = \pi\) (where \(\pi_1 : U \times F \to U\) is projection on the first
factor). Such a homeomorphism is a local trivialization of \(E\) over \(U\).
The space \(E\) is the total space of the bundle, \(M\) is its
base, and \(\pi\) is its projection.
Definition: Smooth Fiber Bundle
If \(E\), \(M\), and \(F\) are smooth manifolds with or without boundary, \(\pi\) is a smooth
map, and the local trivializations can be chosen to be diffeomorphisms, then \(\pi : E \to M\)
is called a smooth fiber bundle.
Definition: Trivial Fiber Bundle and Global Trivialization
A fiber bundle is trivial if it admits a local trivialization over the entire
base. Such a trivialization is called a global trivialization. A smooth fiber
bundle is smoothly trivial if it admits a global trivialization that is a
diffeomorphism.
The definition mirrors that of a vector bundle exactly, with the linear structure on each fiber
replaced by mere homeomorphism with \(F\). What is lost is the algebraic structure. There are no
"sections of a fiber bundle" in the linear sense (one cannot add or scale them, since fibers carry
no operations), and no bundle homomorphisms induced by linear maps on fibers. What is gained is
flexibility. Any topological space can serve as the model fiber, and the theory accommodates
constructions where the natural fibers are not linear.
Four examples of fiber bundles
(a) Product fiber bundles.
For topological spaces \(M\) and \(F\), the product
\(M \times F\) with projection \(\pi_1 : M \times F \to M\) onto the first factor is a fiber
bundle, the product fiber bundle. Its global trivialization is the identity
\(M \times F \to M \times F\). Every product fiber bundle is trivial, and conversely the trivial
fiber bundles are exactly the product bundles up to global trivialization.
(b) Vector bundles as fiber bundles.
Every rank-\(k\) vector bundle is a fiber
bundle with model fiber \(\mathbb{R}^k\). The local trivializations of the vector bundle
satisfy the projection-compatibility condition of fiber-bundle trivializations, and the
additional linear structure on each fiber is the extra data that distinguishes the vector-bundle
case from the general one.
(c) Möbius-band fiber bundle.
Let \(\pi : E \to \mathbb{S}^1\) be the
Möbius bundle,
with quotient map \(q : \mathbb{R}^2 \to E\). The image of \(\mathbb{R} \times [-1, 1]\) under
\(q\) is a closed sub-band of \(E\), and it is a fiber bundle over \(\mathbb{S}^1\) with model
fiber \([-1, 1]\). The same covering-and-quotient local trivializations restrict to give
homeomorphisms with \(U \times [-1, 1]\) on each evenly covered open arc. This fiber bundle is
not trivial. A global trivialization would carry each fiber homeomorphically onto
\(\{p\} \times [-1, 1]\), hence would carry the set of fiber endpoints onto
\(\mathbb{S}^1 \times \{-1, 1\}\). The endpoints of the sub-band form the single circle
\(q(\mathbb{R} \times \{-1, 1\})\), because \((x, 1)\) and \((x + 1, -1)\) are
identified, whereas \(\mathbb{S}^1 \times \{-1, 1\}\) has two components.
(d) Covering maps as fiber bundles.
Every
smooth covering map
\(\pi : E \to M\) is a smooth fiber bundle whose model fiber is a discrete space \(F\) with
one point for each sheet. The number of sheets is constant. Over an evenly covered open set
\(U\), each component of \(\pi^{-1}(U)\) contains exactly one point of each fiber
\(\pi^{-1}(q)\) with \(q \in U\), so the cardinality of \(\pi^{-1}(q)\) is locally constant in
\(q\), hence constant because \(M\) is connected. This cardinality is countable because each
fiber is a discrete subset of the second-countable space \(E\), so \(F\) is a smooth
\(0\)-manifold. For each evenly covered open subset \(U \subseteq M\), choose a bijection
between the components of \(\pi^{-1}(U)\) and \(F\). The diffeomorphism
\(\Phi : \pi^{-1}(U) \to U \times F\) sending each point to its base image and to the point of
\(F\) assigned to its component is a local trivialization. The connection between covering
theory and bundle theory recovers, in the discrete-fiber case, results already established for
covering maps earlier in the manifold series.
The broader picture
Several important classes of fiber bundles arise repeatedly in differential geometry without fitting
the vector-bundle framework. Principal \(G\)-bundles are fiber bundles whose model
fiber is a Lie group \(G\), with \(G\) acting freely and transitively on each fiber by right
multiplication. They are the natural setting for connections and gauge theories.
Associated bundles are built from a principal \(G\)-bundle and a representation of
\(G\) on a model fiber \(F\), producing fiber bundles whose model fiber is \(F\) and whose
structure group is \(G\). A vector bundle is the special case where the representation is on
\(\mathbb{R}^k\) (or \(\mathbb{C}^k\)).
The frame bundle of a vector bundle is a
principal \(GL(k, \mathbb{R})\)-bundle whose fiber at each point is the set of bases of the vector
bundle's fiber there. The Hopf fibration
\(\mathbb{S}^3 \to \mathbb{S}^2\) is a non-trivial fiber bundle with model fiber \(\mathbb{S}^1\),
one of the first historical examples of a non-trivial fiber bundle with non-discrete fibers.
Where the manifold series has arrived
The manifold series began with the question of how to attach calculus to spaces more general
than \(\mathbb{R}^n\). The answer was a hierarchy: topological manifolds, smooth manifolds,
tangent vectors, vector fields, flows, Lie groups and their algebras, submanifolds with their
tangent and normal bundles. Here the arc arrives at the recognition that much of this material
shares a common framework. Tangent vectors are the points of the tangent bundle, vector fields
are its sections, and the tangent and normal bundles of a submanifold are vector bundles over
that submanifold. Many of the further structures still to come, among them differential forms,
Riemannian metrics, and tensor fields, are likewise sections of vector bundles.
The bundle is the right abstraction because it captures both the local product structure that
makes calculus possible and the global topology that determines what is and is not trivial.
Subsequent developments build on this framework, either directly through additional sections
(metrics, differential forms) or by enriching it with extra structure (connections, group
actions, principal bundles). Riemannian geometry, symplectic geometry, complex geometry, gauge
theory, and the bundle-based formulations of equivariant deep learning all proceed this way.