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 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. Vector fields, differential forms, Riemannian metrics, and tensor
fields are all of this kind.
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
Condition (2) above requires a homeomorphism \(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\)
that satisfies \(\pi_U \circ \Phi = \pi\) and restricts to a vector space isomorphism on each
fiber. Such a homeomorphism 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. It is the simplest such example, and its
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
\(\pi^{-1}(U) \to U \times \mathbb{R}\). Two such trivializations built from the lifts
\(\widetilde U + m\) and \(\widetilde U + n\) are related by
\((p, y) \mapsto \bigl(p, (-1)^{m-n} y\bigr)\), so the sign records the parity of the shift
between the lifts. In either case the transition is smooth in \(p\) and linear in \(y\). The
chart lemma developed in the next 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 \(\pi : E \to \mathbb{S}^1\) admits no global trivialization, and is
therefore not a trivial bundle.
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 \(E \setminus q(\mathbb{R} \times \{0\})\). Every class of a point with
\(y \neq 0\) has a representative whose second coordinate is positive, so this complement
is the image under \(q\) of the connected set \(\mathbb{R} \times (0, \infty)\) and is
therefore connected. A global trivialization would be a homeomorphism
\(E \to \mathbb{S}^1 \times \mathbb{R}\) restricting to a linear isomorphism on each
fiber, so it would carry the zero section onto the zero section and restrict to a
homeomorphism between the two complements. Connectedness is a topological invariant, so no
such homeomorphism exists.
A second viewpoint is equivalent to the argument 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,
because going once around the base forces a sign flip. This obstruction is the same connectedness
phenomenon stated differently. The language of sections will be developed later, and 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 page specializes, when applied to \(TM\), to a familiar object built earlier in the
manifold series. The slogan that \(TM\) "looks locally like \(M \times \mathbb{R}^n\)" has been
true since natural coordinates were introduced. It 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. They have the same fibers and
the same local picture, but a 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
Vector spaces over a base, locally identified with \(U \times \mathbb{R}^k\), are the same local
data in both cases. The Möbius bundle showed that such data 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
\[
\Phi \circ \Psi^{-1}(p, v) = \bigl(p,\, \tau(p) v\bigr)
\quad \text{for all } (p, v) \in (U \cap V) \times \mathbb{R}^k .
\]
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
from the \(\varphi\)-trivialization to the \(\psi\)-trivialization 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 \(p \in M\), let \(\alpha\) be any index with \(p \in U_\alpha\) and let
\((V_p, \varphi_p)\) be any smooth chart for \(M\) satisfying
\(p \in V_p \subseteq U_\alpha\). Every admissible choice is retained, so the charts
built below form a collection indexed by all such pairs and not by the points of \(M\)
alone. Write
\(\widehat V_p = \varphi_p(V_p) \subseteq \mathbb{R}^n\) (or \(\mathbb{H}^n\) at boundary
points, where \(n = \dim M\)). Define
\[
\widetilde \varphi_p : \pi^{-1}(V_p) \to \widehat V_p \times \mathbb{R}^k ,
\quad
\widetilde \varphi_p = (\varphi_p \times \mathrm{id}_{\mathbb{R}^k}) \circ \Phi_\alpha\big|_{\pi^{-1}(V_p)} .
\]
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 of all
charts \((\pi^{-1}(V_p),\, \widetilde \varphi_p)\) obtained in this way.
Bijectivity and openness of overlap images.
The former is immediate from the construction. 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.
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
\((z, v) \mapsto (z, \tau_{\alpha\beta}(z) v)\) by hypothesis (3). 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.
Countable subcover.
Because \(M\) is second-countable, countably many of the
\(V_p\) cover \(M\), and the corresponding charts
\(\{(\pi^{-1}(V_p), \widetilde\varphi_p)\}\) cover \(E\).
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.
All five conditions of the smooth manifold chart lemma hold. Thus \(E\) 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 (2). 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 (3),
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 \(\tau\) and in 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 a cover exists because the
pairwise intersections of two open covers again form 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
\[
\begin{align*}
\tau(p) &= \tau'(p) \oplus \tau''(p) \\\\
&= \begin{pmatrix} \tau'(p) & 0 \\\\ 0 & \tau''(p) \end{pmatrix}
\in GL(k' + k'', \mathbb{R}) ,
\end{align*}
\]
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. The most prominent case is the
splitting of the restriction of the tangent bundle to a submanifold into 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 \(S\), 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. The two 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 \(S\). The rank of \(TS\) equals \(\dim S\), while the rank of
\(TM|_S\) equals \(\dim M\). Their relationship is one of the structural payoffs of the framework,
and it is taken up via the language of bundle homomorphisms and subbundles further on in the
manifold series. In particular, \(TM|_S\) splits as \(TS \oplus NS\) when \(S\) is embedded in a
Riemannian manifold, where \(NS\) is the
normal bundle.
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, among them the dual bundle, the tensor product of bundles, the
bundle of \(k\)-th exterior powers, the cotangent bundle, and 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.