Definition and Local Trivializations
The tangent bundle
of a smooth \(n\)-manifold \(M\) 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
\(\Phi|_{E_q} : E_q \to \{q\} \times \mathbb{R}^k \cong \mathbb{R}^k\)
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 (for smooth bundles, 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 continuously with the base point, and smoothly when the
trivializations are smooth local trivializations.
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 can fail. 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. For trivializations built from lifts \(\widetilde U\) of \(U\) and \(\widetilde V\) of
another evenly covered set \(V\), the points of \(\widetilde U\) and \(\widetilde V\) over a common
point of \(U \cap V\) differ by an integer that depends continuously on that point. This integer is
therefore constant on each connected component of \(U \cap V\), and there the transition is
multiplication by the corresponding constant sign. In each 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 from each total space the set consisting of the zero vector of each fiber (the zero
section). 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 \(\mathbb{R}^n\). By construction,
\(\pi_U \circ \Phi = \pi\) 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 \(TM\), up to listing the two blocks of coordinates in the other order when
\((U, \varphi)\) is a boundary chart. 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. The transition functions and
the chart lemma developed below specialize, when applied to \(TM\), to the coordinate Jacobians and
the natural coordinate charts built earlier in the manifold series. The slogan that \(TM\) "looks
locally like \(M \times \mathbb{R}^n\)" has been true since those charts 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 the product and Möbius examples. 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,
or of its
boundary version
when \(M\) has boundary.
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
\[
\widetilde \varphi_p : \pi^{-1}(V_p) \to \widehat V_p \times \mathbb{R}^k ,
\quad
\widetilde \varphi_p = (\varphi_p \times \mathrm{id}_{\mathbb{R}^k}) \circ \Phi_\alpha|_{\pi^{-1}(V_p)} .
\]
Each \(\widetilde \varphi_p\) is a bijection between \(\pi^{-1}(V_p)\) and
\(\widehat V_p \times \mathbb{R}^k\), an open subset of \(\mathbb{R}^{n+k}\) when \((V_p, \varphi_p)\)
is an interior chart, 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 \(\varphi_p \times \mathrm{id}_{\mathbb{R}^k}\). When \((V_p, \varphi_p)\) is a boundary
chart we list the fiber coordinates first, so that the image is \(\mathbb{R}^k \times \widehat V_p\),
an open subset of \(\mathbb{H}^{n+k}\) with the half-space coordinate last. This is the same
reordering used in the
construction of the tangent bundle.
It is a linear isomorphism of \(\mathbb{R}^{n+k}\) and we suppress it from the notation below.
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 the model space \(\mathbb{R}^{n+k}\) or \(\mathbb{H}^{n+k}\) since
\(\varphi_p(V_p \cap V_q)\) is open in \(\mathbb{R}^n\) or \(\mathbb{H}^n\) respectively, the
chart being a homeomorphism onto its image.
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
\[
\widetilde \varphi_p \circ \widetilde \varphi_q^{-1}
= (\varphi_p \times \mathrm{id}_{\mathbb{R}^k})
\circ \bigl(\Phi_\alpha \circ \Phi_\beta^{-1}\bigr)
\circ (\varphi_q \times \mathrm{id}_{\mathbb{R}^k})^{-1} .
\]
The composite has domain \(\varphi_q(V_p \cap V_q) \times \mathbb{R}^k\), and on
\((V_p \cap V_q) \times \mathbb{R}^k\) 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. When \(M\) has
boundary, both factors are smooth in the half-space sense, so the composite extends smoothly
across the hyperplane and is smooth in that sense, which is the smoothness the boundary version
of the chart lemma asks of its transition maps.
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 \(M\),
with or without boundary, 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 of a Riemannian
manifold 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
\(E|_S = \bigsqcup_{p \in S} E_p \subseteq E\), with projection
\(\pi|_{E|_S} : E|_S \to S\). For each local trivialization
\(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) of \(E\) over \(U \subseteq M\), the
restriction \(\Phi|_{\pi^{-1}(U \cap S)} : \pi^{-1}(U \cap S) \to (U \cap S) \times \mathbb{R}^k\)
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 \(M\), with or without boundary, as
recorded below).
When \(E\) is a smooth vector bundle and \(S \subseteq M\) is an
embedded
or immersed submanifold,
or a submanifold with boundary,
\(E|_S\) inherits a smooth vector bundle structure over \(S\) via the chart lemma. The transition
functions \(\tau_{\alpha\beta}\) of a cover \(\{(U_\alpha, \Phi_\alpha)\}\) of \(M\) by smooth
local trivializations of \(E\) restrict to smooth maps on \(U_\alpha \cap U_\beta \cap S\) into
\(GL(k, \mathbb{R})\). Each restricted map is the composition of the smooth inclusion
\(S \hookrightarrow M\) with the original transition function \(\tau_{\alpha\beta}\). The
hypotheses of the chart lemma are met. When \(S\) is immersed but not embedded, the topology that
the chart lemma places on \(E|_S\) need not be the subspace topology inherited from \(E\), just as
the manifold topology of \(S\) need not be its subspace topology in \(M\). One important special
case of restriction deserves its own name.
Definition: Ambient Tangent Bundle
If \(S \subseteq M\) is a smooth (embedded or immersed) submanifold, with or without boundary,
the restriction
\(TM|_S\) 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 \(\dim S = \dim M\).
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 later pages treat it in the language of bundle homomorphisms and subbundles. 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
construction that applies a fixed operation of linear algebra to the fibers over each point.
The operation must also act smoothly on the linear isomorphisms between finite-dimensional
vector spaces, compatibly with composition and identities. Such constructions include the dual
bundle, the tensor product of bundles, the bundle of alternating tensors on the fibers, the
cotangent bundle, and the bundle of mixed tensors of a given type. Each is determined by a rule
that turns the original transition functions into new ones. The rule is the transpose-inverse
for the dual and the Kronecker product for tensor products. For the bundle of alternating
tensors, it is the map that the transpose-inverse induces on the fibers of that bundle. The
chart lemma then assembles the bundle.