Definition and Local Trivializations
The
tangent bundle
of a smooth manifold is a smooth manifold in its own right, and the natural coordinates we built on it
make it look, near every point of the base, like the Cartesian product of an open subset of \(M\) with
\(\mathbb{R}^n\). A whole class of structures shares this local product picture: a collection of vector
spaces, one over each point of a base manifold, assembled in a way that is locally trivial but may be
globally twisted. These are vector bundles, and the
tangent bundle
is the first example. Many of the constructions of smooth manifold theory — vector fields, differential
forms, Riemannian metrics, tensor fields — are most naturally phrased as objects associated to a vector
bundle, so the language pays for itself even when only the tangent bundle is in view.
Definition: Real Vector Bundle
Let \(M\) be a topological space. A (real) vector bundle of rank \(k\) over \(M\)
is a topological space \(E\) together with a surjective continuous map \(\pi : E \to M\) satisfying
the following two conditions:
-
For each \(p \in M\), the fiber \(E_p = \pi^{-1}(p)\) over \(p\) is endowed
with the structure of a \(k\)-dimensional real vector space.
-
For each \(p \in M\), there exist a neighborhood \(U\) of \(p\) in \(M\) and a homeomorphism
\(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) satisfying
\(\pi_U \circ \Phi = \pi\) (where \(\pi_U : U \times \mathbb{R}^k \to U\) is projection on the
first factor), such that for each \(q \in U\) the restriction
\(\Phi\big|_{E_q} : E_q \to \{q\} \times \mathbb{R}^k \cong \mathbb{R}^k\)
is a vector space isomorphism.
The space \(E\) is the total space of the bundle, \(M\) is its base,
and \(\pi\) is its projection. A rank-1 vector bundle is also called a
(real) line bundle.
Definition: Local Trivialization
A homeomorphism \(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) as in condition (ii) above — that
is, one satisfying \(\pi_U \circ \Phi = \pi\) and restricting to a vector space isomorphism on each
fiber — is called a local trivialization of \(E\) over \(U\). The defining
condition of a vector bundle is that local trivializations cover the base.
The two arrows expressed by \(\pi_U \circ \Phi = \pi\) record that a local trivialization preserves the
base point: a vector \(v \in E_p\) lying over \(p\) is sent to a pair \((p, w)\) for some
\(w \in \mathbb{R}^k\), never to a pair over a different base point. The second clause records that the
only freedom in the identification is the choice of basis: on each fiber, \(\Phi\) is a linear
isomorphism, and different trivializations differ by a linear change of basis as one varies over the
fibers (this is the content of the transition function lemma below).
The smooth refinement
Definition: Smooth Vector Bundle
Suppose \(M\) and \(E\) are
smooth manifolds with or without boundary,
and \(\pi : E \to M\) is a smooth map that exhibits \(E\) as a vector bundle over \(M\). If the
local trivializations of \(E\) can be chosen to be diffeomorphisms onto their images, then \(E\) is
called a smooth vector bundle, and any local trivialization that is a
diffeomorphism onto its image is called a smooth local trivialization.
Complex vector bundles are defined identically, with "real vector space" and \(\mathbb{R}^k\) replaced
by "complex vector space" and \(\mathbb{C}^k\). All vector bundles in what follows are real, and the
qualifier will be dropped.
The figure to keep in mind
A local trivialization assigns coordinates to the vectors of \(E\) over the open set \(U\) by
writing each \(v \in \pi^{-1}(U)\) as a pair \((p, w)\) with \(p = \pi(v)\) and
\(w \in \mathbb{R}^k\). The first coordinate records the base point, the second records the vector
in the chosen basis of the fiber. Different choices of trivialization correspond to different
bases of the fibers over \(U\), varying smoothly with the base point.
Triviality
Definition: Trivial Bundle and Global Trivialization
A local trivialization defined on all of \(M\) is called a global trivialization.
A vector bundle that admits a global trivialization is called a trivial bundle.
For a smooth bundle, if the global trivialization is a diffeomorphism the bundle is
smoothly trivial, and the total space is diffeomorphic (not merely homeomorphic)
to \(M \times \mathbb{R}^k\). Throughout this site, "trivial" applied to a smooth bundle means
smoothly trivial unless explicitly stated otherwise.
Proposition: The Projection of a Smooth Vector Bundle Is a Submersion
The projection \(\pi : E \to M\) of a smooth vector bundle is a surjective
smooth submersion.
Proof:
Surjectivity is part of the definition; we show \(\pi\) is a submersion. Let \(v \in E\) and set
\(p = \pi(v)\). Choose a smooth local trivialization
\(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) on a neighborhood \(U\) of \(p\). The projection
on the first factor \(\pi_U : U \times \mathbb{R}^k \to U\) is a smooth submersion (its
differential is surjective at every point), and \(\Phi\) is a diffeomorphism. From
\(\pi = \pi_U \circ \Phi\) on \(\pi^{-1}(U)\), the differential \(d\pi_v\) factors as
\(d(\pi_U)_{\Phi(v)} \circ d\Phi_v\); the first factor is surjective and the second is an
isomorphism, so the composition is surjective.
Examples: Product, Möbius, and Tangent Bundles
Three examples set the conceptual range. The first is the simplest possible vector bundle and shows
what triviality looks like. The second is the simplest non-trivial example, and it shows that
the global product picture genuinely can fail. The third is the example for whose sake the framework
was set up.
The product bundle
For any topological space \(M\) and any \(k \ge 0\), the Cartesian product \(E = M \times \mathbb{R}^k\)
with projection \(\pi_1 : M \times \mathbb{R}^k \to M\) on the first factor is a rank-\(k\) vector
bundle over \(M\), and the identity map \(M \times \mathbb{R}^k \to M \times \mathbb{R}^k\) is a global
trivialization. This is the product bundle, and it is trivial by definition. If \(M\)
is a smooth manifold (with or without boundary), the product bundle is smoothly trivial in the same
way: every fiber inherits the linear structure of \(\mathbb{R}^k\), the projection is smooth, and the
identity is a diffeomorphism.
The Möbius bundle
Triviality is the exception, not the rule. The following construction produces a rank-1 vector bundle
over the circle that is not trivial — the simplest such example, and one whose total space
can be drawn.
Define an equivalence relation on \(\mathbb{R}^2\) by
\[
(x, y) \sim (x', y') \iff (x', y') = (x + n,\, (-1)^n y) \text{ for some } n \in \mathbb{Z} .
\]
Let \(E = \mathbb{R}^2 / \sim\) be the quotient space and \(q : \mathbb{R}^2 \to E\) the quotient map.
Each equivalence class meets the closed strip \([0, 1] \times \mathbb{R}\) in either one interior point
or two points of the form \((0, y)\) and \((1, -y)\) on opposite edges; geometrically, \(E\) is the
strip with its right edge identified to its left edge after a flip in the second coordinate. The image
of \([0, 1] \times [-r, r]\) under \(q\) is the familiar paper Möbius band of half-width \(r\), and
\(E\) is the union of these bands as \(r \to \infty\).
Definition: The Möbius Bundle
Let \(\varepsilon : \mathbb{R} \to \mathbb{S}^1\) be the
smooth covering map
\(\varepsilon(t) = (\cos 2\pi t, \sin 2\pi t)\), and let \(E = \mathbb{R}^2/\sim\) be the quotient
space above. The map \(\varepsilon \circ \pi_1 : \mathbb{R}^2 \to \mathbb{S}^1\) is constant on
equivalence classes (because \(\varepsilon(x + n) = \varepsilon(x)\)), so it descends to a
continuous map \(\pi : E \to \mathbb{S}^1\). The resulting line bundle \(\pi : E \to \mathbb{S}^1\)
is called the Möbius bundle.
The construction makes \(\pi : E \to \mathbb{S}^1\) into a smooth real line bundle: each fiber
\(E_p\) inherits a one-dimensional real vector space structure from the \(y\)-coordinate of any
representative \((x, y)\) (the relation \(y \mapsto (-1)^n y\) is linear, so the structure is
well-defined), and local trivializations are obtained over open arcs of \(\mathbb{S}^1\) by lifting
through \(\varepsilon\). Specifically, if \(U \subseteq \mathbb{S}^1\) is an open subset evenly covered
by \(\varepsilon\) and \(\widetilde U \subseteq \mathbb{R}\) is one component of \(\varepsilon^{-1}(U)\),
then \(q\) restricts to a homeomorphism \(\widetilde U \times \mathbb{R} \to \pi^{-1}(U)\) whose
inverse, post-composed with \(\varepsilon \times \mathrm{id}_{\mathbb{R}}\), gives a local
trivialization \(\pi^{-1}(U) \to U \times \mathbb{R}\). On the overlap of two such trivializations
coming from adjacent lifts \(\widetilde U,\, \widetilde U + 1\), the transition takes the form
\((p, y) \mapsto (p, \pm y)\) with the sign depending on whether the lifts agree or differ by an odd
shift; in both cases the transition is smooth in \(p\) and linear in \(y\). The chart lemma developed
later in this section therefore gives \(E\) a unique smooth structure with respect to which \(\pi\)
is a smooth bundle projection and these trivializations are smooth.
What distinguishes the Möbius bundle from the trivial line bundle \(\mathbb{S}^1 \times \mathbb{R}\)
is the global topology of its total space.
Proposition: The Möbius Bundle Is Not Trivial
The Möbius bundle \(\pi : E \to \mathbb{S}^1\) is not isomorphic to the product line bundle
\(\mathbb{S}^1 \times \mathbb{R} \to \mathbb{S}^1\).
Proof Sketch:
Remove the zero section from each total space. From the product bundle one obtains
\(\mathbb{S}^1 \times (\mathbb{R} \setminus \{0\})\), which has two connected components,
corresponding to the positive and negative half-lines in each fiber. From the Möbius bundle one
obtains \(E \setminus q(\mathbb{R} \times \{0\})\); the equivalence \((x, y) \sim (x + 1, -y)\)
identifies positive and negative half-lines of adjacent fibers, so this complement is connected.
Connectedness is a topological invariant, so the two total spaces are not homeomorphic, and in
particular the bundles are not isomorphic.
A second viewpoint, equivalent to the above: a global trivialization would yield a nowhere-vanishing
global section by composing the constant section \(p \mapsto (p, 1)\) of the product bundle with the
trivializing isomorphism. The Möbius bundle admits no such section — going once around the base
forces a sign flip — and this obstruction is the same connectedness phenomenon stated differently.
The language of sections will be developed later; the connectedness argument suffices for now.
The tangent bundle as a vector bundle
The tangent bundle is the example whose study motivated the entire framework. Everything needed for
its identification as a vector bundle has already been built; this section makes that identification
explicit.
Proposition: The Tangent Bundle Is a Smooth Vector Bundle
Let \(M\) be a smooth \(n\)-manifold with or without boundary. With the standard projection
\(\pi : TM \to M\), the natural vector space structure on each fiber \(T_pM\), and the topology
and smooth structure constructed earlier, \(TM\) is a smooth vector bundle of rank \(n\) over
\(M\).
Proof:
The base \(M\) and total space \(TM\) are smooth manifolds (the latter by the
smooth structure on the tangent bundle),
and the projection \(\pi : TM \to M\) is smooth. Each fiber \(T_pM\) is an \(n\)-dimensional real
vector space by construction, so it remains only to exhibit smooth local trivializations.
Let \((U, \varphi)\) be a smooth chart on \(M\) with coordinate functions \((x^1, \dots, x^n)\).
Define \(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^n\) by recording each tangent vector's base
point and its components in the coordinate basis:
\[
\Phi\!\left( v^i \frac{\partial}{\partial x^i}\bigg|_p \right) = \bigl(p,\, (v^1, \dots, v^n)\bigr) .
\]
On each fiber \(T_pM\), \(\Phi\) is the linear isomorphism sending the coordinate basis
\((\partial/\partial x^i|_p)\) to the standard basis of \(\mathbb{R}^n\); in particular,
\(\pi_U \circ \Phi = \pi\) where \(\pi_U : U \times \mathbb{R}^n \to U\) is projection on the
first factor.
Smoothness of \(\Phi\) reduces to the natural coordinate chart on \(TM\): the composite
\((\varphi \times \mathrm{id}_{\mathbb{R}^n}) \circ \Phi\) on \(\pi^{-1}(U)\) sends
\(v^i \partial/\partial x^i|_p\) to \(\bigl(x^1(p), \dots, x^n(p), v^1, \dots, v^n\bigr)\), which
is exactly the natural coordinate chart \(\widetilde\varphi\) used to build the smooth structure
on \(TM\). Since both \(\widetilde\varphi\) and \(\varphi \times \mathrm{id}_{\mathbb{R}^n}\) are
diffeomorphisms onto their images, so is \(\Phi\). Thus \(\Phi\) is a smooth local trivialization,
and every point of \(M\) lies in the domain of one such.
The tangent bundle is therefore the prototypical smooth vector bundle: every construction in the rest
of this chapter specializes, when applied to \(TM\), to a familiar object built earlier in the
manifold series. The slogan that \(TM\) "looks locally like \(M \times \mathbb{R}^n\)" — true since
natural coordinates were introduced — is now upgraded to the formal statement that the local product
picture is the structure of a smooth vector bundle.
Two questions raised by the examples
The product, Möbius, and tangent bundles together pose the two structural questions developed in
what follows. First, when one assembles vector spaces fiber-by-fiber and chooses local
trivializations, what data on the overlaps is needed to glue them into a vector bundle? The
product and Möbius examples differ only in the gluing — same fibers, same local picture, different
global outcome. Second, when can the local product picture be globalized to an actual product?
For the product bundle the answer is trivially yes; for the Möbius bundle it is no; for the
tangent bundle it depends on the manifold, and a satisfactory answer requires the language of
global sections, taken up on the next page.
Transition Functions and the Chart Lemma
The Möbius bundle showed that the same local data — vector spaces over a base, locally identified with
\(U \times \mathbb{R}^k\) — can be assembled into bundles that are not globally a product. The
structure that records the difference is the way two trivializations are related on the overlap of
their domains. We turn this observation into a lemma, then into a construction principle: starting
from disjoint fibers and prescribed overlap data, one can manufacture a vector bundle.
The transition between two trivializations
Lemma: Transition Between Local Trivializations
Let \(\pi : E \to M\) be a smooth vector bundle of rank \(k\), and let
\(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) and
\(\Psi : \pi^{-1}(V) \to V \times \mathbb{R}^k\) be two smooth local trivializations with
\(U \cap V \neq \emptyset\). There is a unique smooth map
\(\tau : U \cap V \to GL(k, \mathbb{R})\) such that
\[
\Phi \circ \Psi^{-1}(p, v) = \bigl(p,\, \tau(p) v\bigr)
\qquad \text{for all } (p, v) \in (U \cap V) \times \mathbb{R}^k .
\]
Proof:
The composite \(\Phi \circ \Psi^{-1}\) is a diffeomorphism from
\((U \cap V) \times \mathbb{R}^k\) to itself. Because both trivializations satisfy
\(\pi_U \circ \Phi = \pi\) and \(\pi_V \circ \Psi = \pi\), the first coordinate of
\(\Phi \circ \Psi^{-1}(p, v)\) is \(p\); the composite therefore has the form
\((p, v) \mapsto \bigl(p, \sigma(p, v)\bigr)\) for some smooth map
\(\sigma : (U \cap V) \times \mathbb{R}^k \to \mathbb{R}^k\). For each fixed \(p\), the map
\(v \mapsto \sigma(p, v)\) is the composition of the fiberwise isomorphisms \(\Psi^{-1}|_{\{p\} \times \mathbb{R}^k}\)
and \(\Phi|_{E_p}\), so it is a linear isomorphism of \(\mathbb{R}^k\); call its matrix
\(\tau(p) \in GL(k, \mathbb{R})\). Thus \(\sigma(p, v) = \tau(p) v\) and
\(\Phi \circ \Psi^{-1}(p, v) = (p, \tau(p) v)\), with \(\tau\) uniquely determined.
It remains to show that \(\tau : U \cap V \to GL(k, \mathbb{R})\) is smooth. Working in any
smooth chart on \(U \cap V\), the entries of the matrix \(\tau(p)\) are recovered by applying
\(\sigma(p, \cdot\,)\) to the standard basis vectors \(e_1, \dots, e_k \in \mathbb{R}^k\): the
\(i\)-th column of \(\tau(p)\) is \(\sigma(p, e_i)\). Each \(\sigma(p, e_i)\) is smooth in \(p\)
as the composition of the smooth map \(p \mapsto (p, e_i)\) with the smooth map \(\sigma\), so
every entry of \(\tau(p)\) is smooth in \(p\). Identifying \(GL(k, \mathbb{R})\) with its
embedding as an open subset of the space of \(k \times k\) matrices, smoothness of all entries
is smoothness of \(\tau\) into \(GL(k, \mathbb{R})\).
Definition: Transition Function
The map \(\tau : U \cap V \to GL(k, \mathbb{R})\) of the preceding lemma is called the
transition function from \(\Psi\) to \(\Phi\). When the local trivializations
come from a cover \(\{(U_\alpha, \Phi_\alpha)\}\) of \(M\), the resulting family of transition
functions is denoted \(\tau_{\alpha\beta} : U_\alpha \cap U_\beta \to GL(k, \mathbb{R})\), with
the convention \(\Phi_\alpha \circ \Phi_\beta^{-1}(p, v) = \bigl(p, \tau_{\alpha\beta}(p) v\bigr)\).
Two consistency identities are immediate from the definition: on triple overlaps
\(U_\alpha \cap U_\beta \cap U_\gamma\) one has \(\tau_{\alpha\beta} \tau_{\beta\gamma} = \tau_{\alpha\gamma}\)
(the cocycle condition), and in particular \(\tau_{\alpha\alpha} = I\) and
\(\tau_{\beta\alpha} = \tau_{\alpha\beta}^{-1}\). The cocycle condition is what makes the assembly of
fibers into a single bundle internally consistent on triple intersections, and it is the input
structure for the chart lemma below.
For the tangent bundle, the transition functions associated with two coordinate charts \((U, \varphi)\)
and \((V, \psi)\) on a smooth manifold are recognizable from work already done: a tangent vector
\(v^i \partial/\partial x^i|_p\) in the \(\varphi\)-chart becomes
\(\tilde v^j \partial/\partial \tilde x^j|_p\) in the \(\psi\)-chart, where the
component transformation
rewrites \(\tilde v^j = (\partial \tilde x^j / \partial x^i)(p)\, v^i\). The transition function is
therefore the Jacobian of the coordinate change,
\(\tau(p) = \bigl[\partial \tilde x^j / \partial x^i(p)\bigr]_{j,i}\), a smooth map
\(U \cap V \to GL(n, \mathbb{R})\). The tangent bundle is the smooth vector bundle whose transition
functions are the coordinate Jacobians.
The vector bundle chart lemma
The transition functions of a vector bundle are an output of the bundle structure. The next result
runs the construction backward: given fibers, an open cover, fiber-to-\(\mathbb{R}^k\) bijections,
and smooth transition data on overlaps, the bundle structure can be reconstructed. This parallels
the
smooth manifold chart lemma,
which built a smooth manifold from chart data without requiring a prior topology, and the proof
reduces to that lemma applied to the total space.
Lemma: Vector Bundle Chart Lemma
Let \(M\) be a smooth manifold with or without boundary. Suppose that for each \(p \in M\) we are
given a real vector space \(E_p\) of some fixed dimension \(k\); let
\(E = \bigsqcup_{p \in M} E_p\) and let \(\pi : E \to M\) be the map sending each element of
\(E_p\) to \(p\). Suppose furthermore that the following data are given:
-
an open cover \(\{U_\alpha\}_{\alpha \in A}\) of \(M\);
-
for each \(\alpha \in A\), a bijection
\(\Phi_\alpha : \pi^{-1}(U_\alpha) \to U_\alpha \times \mathbb{R}^k\) whose restriction to
each \(E_p\) is a vector space isomorphism \(E_p \to \{p\} \times \mathbb{R}^k \cong \mathbb{R}^k\);
-
for each \(\alpha, \beta \in A\) with \(U_\alpha \cap U_\beta \neq \emptyset\), a smooth map
\(\tau_{\alpha\beta} : U_\alpha \cap U_\beta \to GL(k, \mathbb{R})\) such that
\(\Phi_\alpha \circ \Phi_\beta^{-1}(p, v) = \bigl(p,\, \tau_{\alpha\beta}(p) v\bigr)\) on
\((U_\alpha \cap U_\beta) \times \mathbb{R}^k\).
Then \(E\) has a unique topology and smooth structure making it a smooth manifold (with or without
boundary) and a smooth rank-\(k\) vector bundle over \(M\), with \(\pi\) as projection and
\(\{(U_\alpha, \Phi_\alpha)\}\) as smooth local trivializations.
Proof:
We construct charts on \(E\) and verify the conditions of the
smooth manifold chart lemma.
For each \(p \in M\), choose an index \(\alpha\) with \(p \in U_\alpha\) and a smooth chart
\((V_p, \varphi_p)\) for \(M\) satisfying \(p \in V_p \subseteq U_\alpha\); write
\(\widehat V_p = \varphi_p(V_p) \subseteq \mathbb{R}^n\) (or \(\mathbb{H}^n\) at boundary points,
where \(n = \dim M\)). Define
\[
\widetilde \varphi_p : \pi^{-1}(V_p) \to \widehat V_p \times \mathbb{R}^k ,
\qquad
\widetilde \varphi_p = (\varphi_p \times \mathrm{id}_{\mathbb{R}^k}) \circ \Phi_\alpha\big|_{\pi^{-1}(V_p)} .
\]
Each \(\widetilde \varphi_p\) is a bijection between \(\pi^{-1}(V_p)\) and the open subset
\(\widehat V_p \times \mathbb{R}^k\) of \(\mathbb{R}^{n+k}\) (or \(\mathbb{H}^{n+k}\)), being the
composition of the bijection \(\Phi_\alpha|_{\pi^{-1}(V_p)} : \pi^{-1}(V_p) \to V_p \times \mathbb{R}^k\)
with the bijection \(\varphi_p \times \mathrm{id}_{\mathbb{R}^k}\).
We verify the five conditions of the smooth manifold chart lemma for the collection
\(\bigl\{(\pi^{-1}(V_p),\, \widetilde \varphi_p) : p \in M\bigr\}\).
Conditions (i)–(ii) (bijectivity and openness of overlap images): bijectivity is
just stated. For two such charts \(\widetilde\varphi_p\) and \(\widetilde\varphi_q\), the image
\(\widetilde\varphi_p\bigl(\pi^{-1}(V_p) \cap \pi^{-1}(V_q)\bigr)\) equals
\(\varphi_p(V_p \cap V_q) \times \mathbb{R}^k\) (because \(\Phi_\alpha\) preserves base points),
which is open in \(\mathbb{R}^{n+k}\) since \(\varphi_p(V_p \cap V_q)\) is open in
\(\mathbb{R}^n\) (or \(\mathbb{H}^n\)) by the smoothness of the chart.
Condition (iii) (smooth compatibility of overlap maps): suppose
\((V_p, \varphi_p)\) comes from index \(\alpha\) and \((V_q, \varphi_q)\) from index \(\beta\),
so on \(V_p \cap V_q \subseteq U_\alpha \cap U_\beta\) the transition function
\(\tau_{\alpha\beta}\) is defined. Compute
\[
\widetilde \varphi_p \circ \widetilde \varphi_q^{-1}
= (\varphi_p \times \mathrm{id}_{\mathbb{R}^k})
\circ \bigl(\Phi_\alpha \circ \Phi_\beta^{-1}\bigr)
\circ (\varphi_q \times \mathrm{id}_{\mathbb{R}^k})^{-1} .
\]
On its domain \(\varphi_q(V_p \cap V_q) \times \mathbb{R}^k\), the middle factor is
\((p, v) \mapsto (p, \tau_{\alpha\beta}(p) v)\) by hypothesis (iii); composing with the smooth
chart-transition map \(\varphi_p \circ \varphi_q^{-1}\) on the first coordinate and applying the
smooth map \(\tau_{\alpha\beta}\) (pulled back through \(\varphi_q^{-1}\)) to the second gives a
smooth map onto \(\varphi_p(V_p \cap V_q) \times \mathbb{R}^k\), with smooth inverse obtained by
the same construction with \(p, q\) and \(\alpha, \beta\) swapped.
Condition (iv) (countable subcover): the cover \(\{V_p : p \in M\}\) of \(M\) has
a countable subcover because \(M\) is second-countable, and the corresponding charts
\(\{(\pi^{-1}(V_p), \widetilde\varphi_p)\}\) cover \(E\).
Condition (v) (Hausdorff separation by chart pairs): given distinct
\(\xi, \eta \in E\), if \(\pi(\xi) = \pi(\eta) = p\) then both lie in the chart \(\pi^{-1}(V_p)\)
for any choice of \(V_p\), and \(\widetilde\varphi_p\) sends them to distinct points of
\(\widehat V_p \times \mathbb{R}^k\) (being a bijection). If \(\pi(\xi) = p \neq q = \pi(\eta)\),
choose disjoint chart neighborhoods \(V_p, V_q\) in \(M\) (using Hausdorffness of \(M\)); then
\(\pi^{-1}(V_p)\) and \(\pi^{-1}(V_q)\) are disjoint chart neighborhoods containing \(\xi\) and
\(\eta\) respectively.
Conditions (i)–(v) of the smooth manifold chart lemma hold; \(E\) acquires a unique topology and
smooth manifold structure (with or without boundary) for which each
\((\pi^{-1}(V_p), \widetilde\varphi_p)\) is a smooth chart. With respect to this structure, each
\(\Phi_\alpha\) is a diffeomorphism: its coordinate representation in the charts
\((\pi^{-1}(V_p), \widetilde\varphi_p)\) on \(E\) and
\((V_p \times \mathbb{R}^k, \varphi_p \times \mathrm{id}_{\mathbb{R}^k})\) on
\(U_\alpha \times \mathbb{R}^k\) is the identity. The projection \(\pi\) is smooth because its
coordinate representation in the chart \((V_p, \varphi_p)\) on \(M\) and
\((\pi^{-1}(V_p), \widetilde\varphi_p)\) on \(E\) is the projection
\((x, v) \mapsto x\), itself smooth. Linearity of \(\Phi_\alpha\) on fibers was assumed in (ii).
Thus the \(\Phi_\alpha\) are smooth local trivializations, and \(\pi : E \to M\) is a smooth rank-\(k\)
vector bundle.
Uniqueness: any smooth structure on \(E\) making the \(\Phi_\alpha\) diffeomorphisms onto their
images must include all the charts \((\pi^{-1}(V_p), \widetilde\varphi_p)\) (they are
compositions of the \(\Phi_\alpha\) with smooth charts on \(M\)), and so coincides with the
structure just constructed.
The chart lemma as recipe
The lemma's hypotheses are exactly what one usually has when assembling a vector bundle from
geometric or algebraic data: a base manifold, a vector space attached to each point, and a
prescription for identifying neighbouring fibers. The cocycle condition (implicit in (iii),
because the \(\Phi_\alpha\) are bijections whose composites must agree on triple overlaps) is the
consistency check. The construction packaged this way decouples the choice of overlap data
\(\tau_{\alpha\beta}\) from the choice of fibers, making explicit how different bundles over the
same base arise from different gluing rules — the trivial line bundle and the Möbius bundle
over \(\mathbb{S}^1\) differ in their \(\tau\), nothing else.
New Bundles from Old
The chart lemma is a construction principle: any time fibers, fiber-to-\(\mathbb{R}^k\) bijections,
and smooth transition data are produced, a smooth vector bundle results. Two such constructions occur
so frequently that they deserve their own names. The first combines two bundles over the same base
into a single bundle whose fiber is the direct sum of the original fibers; the second restricts a
bundle to a subset of the base.
The Whitney sum
Definition: Whitney Sum of Vector Bundles
Let \(E' \to M\) and \(E'' \to M\) be smooth vector bundles over the same smooth manifold \(M\),
of ranks \(k'\) and \(k''\). The Whitney sum of \(E'\) and \(E''\), denoted
\(E' \oplus E''\), is the smooth vector bundle over \(M\) of rank \(k' + k''\) with fiber
\((E' \oplus E'')_p = E'_p \oplus E''_p\) at each point \(p \in M\) (the direct sum of the
original fibers as real vector spaces). Its total space is the disjoint union
\(\bigsqcup_{p \in M} (E'_p \oplus E''_p)\) with the obvious projection.
The bundle structure on \(E' \oplus E''\) is constructed by the chart lemma. Cover \(M\) by open sets
\(U\) over which both \(E'\) and \(E''\) admit smooth local trivializations
\(\Phi' : (\pi')^{-1}(U) \to U \times \mathbb{R}^{k'}\) and
\(\Phi'' : (\pi'')^{-1}(U) \to U \times \mathbb{R}^{k''}\) (such common refinements exist because the
intersection of two open covers is an open cover). For a fiber element
\((v', v'') \in E'_p \oplus E''_p\), write \(\Phi'(v') = (p, w')\) and \(\Phi''(v'') = (p, w'')\), and
define \(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^{k' + k''}\) by
\[
\Phi(v', v'') = \bigl(p,\, (w', w'')\bigr) ,
\]
which is a bijection and a fiberwise isomorphism. Given two such pairs of trivializations over \(U\)
and \(\widetilde U\), with transition functions
\(\tau' : U \cap \widetilde U \to GL(k', \mathbb{R})\) and
\(\tau'' : U \cap \widetilde U \to GL(k'', \mathbb{R})\), the transition function for
\(E' \oplus E''\) is
\[
\tau(p) = \tau'(p) \oplus \tau''(p) = \begin{pmatrix} \tau'(p) & 0 \\ 0 & \tau''(p) \end{pmatrix}
\in GL(k' + k'', \mathbb{R}) ,
\]
smooth as a block-diagonal matrix-valued function of \(p\) because each block is. The vector bundle
chart lemma applies and produces the smooth structure on \(E' \oplus E''\).
The block-diagonal form is the structural content: the Whitney sum carries no interaction between
its two summands, only the parallel combination of their gluings. Whitney sums will appear when
decomposing a vector bundle along a complementary direction — most prominently when the restriction
of the tangent bundle to a submanifold splits as the tangent bundle of the submanifold plus the
normal bundle, a fact taken up further on in the manifold series.
Restriction
Restricting a bundle to a subset of the base requires nothing new: the fibers over the subset are
already vector spaces, and the local trivializations of the original bundle restrict to local
trivializations of the new one.
Definition: Restriction of a Vector Bundle
Let \(\pi : E \to M\) be a rank-\(k\) vector bundle and \(S \subseteq M\) any subset. The
restriction of \(E\) to \(S\) is the set
\(E\big|_S = \bigsqcup_{p \in S} E_p \subseteq E\), with projection \(\pi\big|_{E|_S} : E|_S \to S\).
For each local trivialization \(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) of \(E\) over
\(U \subseteq M\), the restriction
\(\Phi\big|_{(\pi|_S)^{-1}(U \cap S)} : (\pi|_S)^{-1}(U \cap S) \to (U \cap S) \times \mathbb{R}^k\)
is a local trivialization of \(E|_S\) over \(U \cap S\). These restricted trivializations cover
\(S\) and exhibit \(E|_S\) as a rank-\(k\) vector bundle over \(S\) (topological in general,
smooth when \(S\) is a smooth submanifold of \(M\), as recorded below).
When \(E\) is a smooth vector bundle and \(S \subseteq M\) is an
embedded
or
immersed submanifold,
\(E|_S\) inherits a smooth vector bundle structure over \(S\) via the chart lemma. The transition
functions of \(E\) restrict to smooth maps on \(U \cap S\) into \(GL(k, \mathbb{R})\) (composition of
the inclusion \(S \hookrightarrow M\), which is smooth, with the original transition function
\(\tau_{\alpha\beta}\)), and the hypotheses of the chart lemma are met. One important special case
deserves its own name.
Definition: Ambient Tangent Bundle
If \(S \subseteq M\) is a smooth (embedded or immersed) submanifold, the restriction
\(TM\big|_S\) of the tangent bundle of \(M\) to \(S\) is called the ambient tangent
bundle over \(M\) (named for the ambient manifold whose tangent bundle is being
restricted; the resulting bundle is a vector bundle over \(S\)). Its fiber at a point
\(p \in S\) is the tangent space \(T_pM\) of the ambient manifold, not the tangent space
\(T_pS\) of the submanifold — these differ unless \(S\) is open in \(M\).
The ambient tangent bundle \(TM|_S\) and the submanifold tangent bundle \(TS\) are two distinct
vector bundles over the same base \(S\): \(TS\) has rank equal to \(\dim S\), while \(TM|_S\) has
rank equal to \(\dim M\). Their relationship — and in particular the splitting
\(TM|_S \cong TS \oplus NS\) when \(S\) is embedded in a Riemannian manifold, where \(NS\) is the
normal bundle —
is one of the structural payoffs of the framework, and is taken up via the language of bundle
homomorphisms and subbundles further on in the manifold series.
Two operations, both via the chart lemma
The Whitney sum and the restriction look like different operations, but the chart lemma reduces
both to specifying transition functions on a common open cover: block-diagonal in the first
case, restricted from the ambient cover in the second. The pattern repeats for every standard
bundle construction — the dual bundle, the tensor product of bundles, the bundle of \(k\)-th
exterior powers, the cotangent bundle, the bundle of \((r,s)\)-tensors — each is determined by a
rule that turns the original transition functions into new ones (transpose-inverse for the dual,
Kronecker product for tensor products, exterior power for forms). The chart lemma then assembles
the bundle. Stating these constructions, and verifying their transition functions, will be the
organizing problem of later developments built on the vector-bundle framework.