Bundle Homomorphisms
A
vector bundle
is a manifold equipped with a fiber structure, so the natural maps between vector bundles must
respect both manifold structure (continuity or smoothness) and fiber structure (linearity, base
point assignment). The result is the class of bundle homomorphisms — maps that 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 \(F\) is also a diffeomorphism it is a
smooth bundle isomorphism, and the bundles are said to be
(smoothly) isomorphic. In the smooth setting, one need not verify smoothness of
the inverse separately: bijectivity plus smoothness of \(F\) is enough.
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 over \(M\).
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 neighbourhood of each point of \(E'\). 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 neighbourhood
\(U\) of \(p\) (such a common \(U\) exists because both bundles are trivializable in a
neighbourhood of \(p\); 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 fibrewise 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 \(M\) (it satisfies \(\pi_M \circ \Phi = \pi\)
and restricts to a vector space isomorphism on each fiber) that is also a diffeomorphism, hence
a smooth bundle isomorphism. 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 \(dF\) was established when the tangent bundle was constructed: in any pair of
smooth charts, the coordinate representation of \(dF\) is the Jacobian matrix of \(F\), itself
smooth. The base relation \(\pi_N \circ dF = F \circ \pi_M\) holds because \(dF\) sends \(T_pM\)
to \(T_{F(p)}N\), and the restriction \(dF_p : T_pM \to T_{F(p)}N\) is a linear map by
construction.
For 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 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 \(C^\infty(M)\) — multiplication
by a base function commutes with the operation. This stronger linearity property is, in fact, not
just a consequence of the bundle-homomorphism origin but a complete characterization: every
\(C^\infty(M)\)-linear map between the
spaces of smooth sections
arises from a bundle homomorphism. This characterization is the algebraic content of the entire
section calculus: working with bundle 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
\[
\widetilde F(\sigma)(p) = F(\sigma(p)) , \qquad p \in M ,
\]
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 \(M\) commutes with \(F\) (because at each \(p\) it is
multiplication by the scalar \(u(p)\), which commutes with the linear map \(F|_{E_p}\)), giving the
stronger property below.
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, 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
\(\sigma_1, \sigma_2 \in \Gamma(E)\) agree on some open set \(U \subseteq M\), then
\(\mathcal{F}(\sigma_1)\) and \(\mathcal{F}(\sigma_2)\) 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
\(\sigma_1, \sigma_2 \in \Gamma(E)\) agree at a single point \(p\), then
\(\mathcal{F}(\sigma_1)(p) = \mathcal{F}(\sigma_2)(p)\). Again it suffices to show that if
\(\tau \in \Gamma(E)\) satisfies \(\tau(p) = 0\), then \(\mathcal{F}(\tau)(p) = 0\). Choose a
smooth local frame \((\sigma_1, \dots, \sigma_k)\) for \(E\) on some neighbourhood \(W\) 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 neighbourhood of \(p\); shrinking that neighbourhood if necessary, we may
assume \(\tau = \widetilde u^i \widetilde \sigma_i\) holds on the neighbourhood by Step 1. Then
by Step 1 again 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 neighbourhood of each point \(v_0 \in E\). Let \(p = \pi(v_0)\), and choose a smooth
local frame \((\sigma_i)\) for \(E\) on a neighbourhood \(W\) of \(p\) and a smooth local frame
\((\sigma'_j)\) for \(E'\) on a (possibly smaller) common neighbourhood; using the extension
lemma, replace \((\sigma_i)\) by global smooth sections \((\widetilde \sigma_i)\) agreeing with
\((\sigma_i)\) on some neighbourhood \(U \subseteq W\) of \(p\), and analogously for the
\(\sigma'_j\). On \(U\), Step 1 gives
\(\mathcal{F}(\widetilde \sigma_i)|_U = \mathcal{F}(\sigma_i)|_U\), and 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
neighbourhood 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. A Riemannian metric on \(M\) is defined as a
\(C^\infty(M)\)-bilinear map
\(g : \Gamma(TM) \times \Gamma(TM) \to C^\infty(M)\), and the lemma's content is that this
algebraic definition automatically descends to a fiber-by-fiber inner product on each \(T_pM\)
(a bundle homomorphism \(TM \otimes TM \to \mathbb{R}\)) — no separate verification needed. The
same applies to tensor fields, almost complex structures, and symplectic forms when defined via
section calculus: \(C^\infty(M)\)-linearity is exactly the algebraic condition that distinguishes
such "tensorial" objects from operators that depend on the global behaviour of sections (such
as differentiation, which is \(\mathbb{R}\)-linear but not \(C^\infty(M)\)-linear because of
the Leibniz rule). Connections fall on the non-tensorial side of this divide — they satisfy a
Leibniz rule and are not bundle homomorphisms — and are organised by a separate algebraic
framework. 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.
A note on notation: by virtue of the characterization, 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.
Differential operators that involve derivation — most prominently the
Lie derivative
\(\mathscr{L}_X : \mathfrak{X}(M) \to \mathfrak{X}(M)\) — 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 \(u \in C^\infty(M)\)).
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 — being a vector
bundle in its own right, and sitting inside the ambient bundle as a topological vector subspace.
The characterization lemma of the previous section produces subbundles from bundle homomorphisms
(kernels and images, treated below); 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 vector 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 of \(E\) (the
inclusion of an embedded 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 \(\mathrm{Id}_M\). Fiber-wise,
\(\iota|_{D_p} : D_p \hookrightarrow E_p\) 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 neighbourhood 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 neighbourhood \(U\) 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 \(q \in U\).
Proof:
Necessity. Suppose \(D\) is a smooth subbundle of \(E\). For each \(p \in M\),
\(D\) admits a smooth local trivialization over some neighbourhood \(U\) 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 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 neighbourhood on which smooth local sections
\(\sigma_1, \dots, \sigma_m\) of \(E\) form a basis of \(D_q\) for each \(q \in U\). By the
completion of local frames
(applied to \(E\) and the partial frame \((\sigma_1, \dots, \sigma_m)\), shrinking \(U\) if
necessary), there exist 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 \(U\). Let
\(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) be the smooth local trivialization of \(E\)
associated with this frame, as given by the
frame ↔ 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 an embedded submanifold (with or
without boundary) of \(U \times \mathbb{R}^k\), so \(D \cap \pi^{-1}(U)\) is an embedded
submanifold of \(\pi^{-1}(U) \subseteq E\). The collection of such neighbourhoods covers \(E\),
so \(D\) is an embedded submanifold 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 the embedding
\(D \cap \pi^{-1}(U) \hookrightarrow U \times \mathbb{R}^m \times \{0\}^{k-m}\) (from the
argument above) with projection onto the first \(m + \dim M\) coordinates, both of which are
diffeomorphisms onto their images. Hence \(\Psi\) is a diffeomorphism, fiber-wise linear, and
satisfies \(\pi_U \circ \Psi = \pi|_D\). Therefore \(\Psi\) is a smooth local trivialization of
\(D\) over \(U\); since every point of \(M\) lies in such a \(U\), \(D\) inherits a smooth
vector bundle structure of rank \(m\) over \(M\), and the inclusion \(D \hookrightarrow E\) is
the inclusion of an embedded submanifold. 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 \((E_1, \dots, E_k)\). For \(0 \le m \le k\), the union
\(D \subseteq E\) with fiber \(D_p = \mathrm{span}(E_1|_p, \dots, E_m|_p)\) is a smooth rank-\(m\)
subbundle of \(E\); the local-section condition holds globally with
\(\sigma_i = E_i\) for \(i = 1, \dots, m\). When \(M = G\) is a Lie group and
\((E_1, \dots, E_n)\) 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 and
\(S \subseteq M\) an immersed \(k\)-submanifold. The
tangent bundle
\(TS\) sits inside the
ambient tangent bundle
\(TM|_S\) as a smooth rank-\(k\) subbundle. To see this, choose a smooth chart \((V, \varphi)\)
on \(S\) and a smooth chart \((W, \psi)\) on \(M\) compatible with the immersion (so that
\(S \cap W\) is parametrized by the first \(k\) coordinates of \(\psi\)). The coordinate vector
fields \(\partial/\partial \varphi^i|_q\) on \(S\) extend to smooth local sections of \(TM|_S\)
on \(S \cap W\), and they span \(T_qS\) at each \(q\); 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 the dimension of the kernel and the
rank of the image are not, in general, constant along the base — they can jump. 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 fiber-wise 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 \(M\), and let
\(F : E \to E'\) be a smooth bundle homomorphism over \(M\). Define
\[
\mathrm{Ker}\, F = \bigcup_{p \in M} \mathrm{Ker}(F|_{E_p}) \subseteq E,
\qquad
\mathrm{Im}\, F = \bigcup_{p \in M} \mathrm{Im}(F|_{E_p}) \subseteq E' .
\]
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 \(r\); let \(k\) and \(k'\) denote
the ranks of \(E\) and \(E'\), 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 neighbourhood \(U\) 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 exactly \(r\) of them are linearly independent at \(p\) (the rank of
\(F|_{E_p}\) being \(r\)). By relabelling 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 neighbourhood
\(U_0 \subseteq U\) 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. The subbundle
\(V \subseteq E|_{U_0}\) spanned by \((\sigma_1, \dots, \sigma_r)\) (by the local-frame
criterion, taking these sections themselves as the local frame on \(U_0\)) is a smooth rank-\(r\)
subbundle of \(E|_{U_0}\) 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
\[
\Psi : E|_{U_0} \to E|_{U_0}, \qquad
\Psi(v) = v - (F|_V)^{-1}\bigl(F(v)\bigr) .
\]
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
\(\Psi(v) = v - (F|_V)^{-1}(0) = v\); \(\Psi\) 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
\(V \oplus \mathrm{Ker}\, F\) recover \(E|_{U_0}\) 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 neighbourhood 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 neighbourhood of \(p\).
Proof:
Fix \(p \in M\), and let \((X_1, \dots, X_k)\) be a smooth local frame for \(D\) over some
neighbourhood \(V\) of \(p\) in \(M\). Because immersed submanifolds are locally embedded, we
may shrink \(V\) so that it is a single slice in some coordinate ball or half-ball
\(U \subseteq \mathbb{R}^n\); the closedness of \(V\) in \(U\) (after this shrinking) allows
the
completion of local frames
to be applied: we can extend \((X_1, \dots, X_k)\) to a smooth local frame
\((\widetilde X_1, \dots, \widetilde X_n)\) for \(T\mathbb{R}^n\) over \(U\). 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
\(\mathrm{span}(E_1|_p, \dots, E_k|_p) = \mathrm{span}(X_1|_p, \dots, X_k|_p) = D_p\) for every
\(p \in V\) (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 \(p \in V\), the vectors \(E_{k+1}|_p, \dots, E_n|_p\) are orthonormal and
orthogonal to every \(E_i|_p\) (\(i \le k\)), hence to all of \(D_p\), so they lie in
\(D_p^\perp\); and they span an \((n-k)\)-dimensional subspace, which by dimension count is all
of \(D_p^\perp\). 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 as the orthogonal complement of \(TM\) inside \(T\mathbb{R}^n|_M\).
Within the present framework, it 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 neighbourhood 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 neighbourhoods 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 kernel and image
sets — defined fibrewise — 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 (as in the orthogonal-complement
construction below, where the homomorphism is projection along a complementary subbundle), 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 on the
next page gives the normal bundle as a tangible geometric object — the same normal bundle that
supplied the tubular neighbourhood theorem — recognized now as one instance of the
orthogonal-complement 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 that fall
outside the vector-bundle framework — covering spaces, principal bundles, frame bundles,
associated bundles, and Hopf fibrations among them — fit naturally into this larger picture, and
several recur in later parts of the site. 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 neighbourhood \(U\) 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 — the same connectedness argument that distinguished the Möbius bundle from the
trivial line bundle applies: removing the central circle \(q(\mathbb{R} \times \{0\})\) leaves
a connected complement, whereas the corresponding complement in the trivial bundle
\(\mathbb{S}^1 \times [-1, 1]\) 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 of the
appropriate cardinality (the number of sheets). For each evenly covered open subset
\(U \subseteq M\) with components of \(\pi^{-1}(U)\) labelled by a discrete set \(S\), the
homeomorphism \(\Phi : \pi^{-1}(U) \to U \times S\) sending each point to its base image and its
component label 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.
The end of the manifold series
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. The arc closes here with the recognition that all of these
constructions — and many of the further structures of differential forms, Riemannian metrics,
and tensor fields still to come — share a common framework: each lives on, or is a section of,
a vector bundle (or, more generally, a fiber bundle) over the manifold. 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 —
Riemannian geometry, symplectic geometry, complex geometry, gauge theory, equivariant deep
learning — build on this framework, either directly through additional sections (metrics,
differential forms) or by enriching it with extra structure (connections, group actions,
principal bundles).