Sections of a Vector Bundle
A
vector bundle
\(\pi : E \to M\) is a family of vector spaces parametrized by the base manifold. The natural objects
living on such a structure are not points of \(E\), but choices of a vector in each fiber, varying
continuously or smoothly with the base point — that is, right inverses of the projection \(\pi\).
For the
tangent bundle,
such choices are precisely vector fields; for a general bundle they are sections, and they will be
the algebraic substrate for everything built on top of vector bundles in later parts of the site —
Riemannian metrics, tensor fields, differential forms, connections.
Definition: Section, Local Section, Global Section
Let \(\pi : E \to M\) be a vector bundle. A section of \(E\) (sometimes called a
cross section) is a continuous map \(\sigma : M \to E\) satisfying
\(\pi \circ \sigma = \mathrm{Id}_M\); equivalently, \(\sigma(p) \in E_p\) for every \(p \in M\).
More generally, a local section of \(E\) is a continuous map
\(\sigma : U \to E\) defined on some open subset \(U \subseteq M\) and satisfying
\(\pi \circ \sigma = \mathrm{Id}_U\); to emphasize the distinction, a section defined on all of
\(M\) is called a global section. A "section" without further qualification
always means a continuous section.
A local section of \(E\) over \(U\) is the same as a global section of the
restricted bundle
\(E|_U\), so the local case reduces formally to the global one over a smaller base. The two terms
coexist because both perspectives — extending a local construction to a global one, or restricting a
global object to a neighbourhood — arise frequently.
Definition: Smooth Section
When \(M\) is a smooth manifold (with or without boundary) and \(E\) is a smooth vector bundle, a
smooth (local or global) section of \(E\) is a section that is smooth as a map
between smooth manifolds.
Allowing the failure of continuity is occasionally useful — for instance, when constructing a section
by an explicit formula on each chart and only later verifying that the pieces fit. The next definition
isolates this looser notion.
Definition: Rough Section
A rough (local or global) section of \(E\) over \(U \subseteq M\) is a map
\(\sigma : U \to E\) (not necessarily continuous) satisfying \(\pi \circ \sigma = \mathrm{Id}_U\).
The zero section
Every vector bundle carries a distinguished global section: at each point, take the zero vector of
the fiber. This is the only section guaranteed to exist on every bundle without further hypothesis,
and its global behaviour is the simplest possible.
Definition: The Zero Section
The zero section of a vector bundle \(\pi : E \to M\) is the global section
\(\zeta : M \to E\) defined by \(\zeta(p) = 0 \in E_p\) for each \(p \in M\). The
support of an arbitrary section \(\sigma\) is the closure of the set
\(\{p \in M : \sigma(p) \neq 0\}\); the zero section is the unique section with empty support.
Proposition: The Zero Section Is Smooth
The zero section of every vector bundle is continuous, and the zero section of every smooth
vector bundle is smooth.
Proof:
Choose a (smooth) local trivialization \(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\). On
\(U\), the composite \(\Phi \circ \zeta\) sends \(p\) to \((p, 0) \in U \times \mathbb{R}^k\),
which is continuous (and smooth, when \(\Phi\) is a diffeomorphism). Because \(\Phi\) is a
homeomorphism (resp. diffeomorphism) onto its image, \(\zeta = \Phi^{-1} \circ (\Phi \circ \zeta)\)
is continuous (resp. smooth) on \(U\); since this holds on a neighbourhood of every point, the
global zero section has the asserted regularity.
Sections of three particular bundles deserve their own names; together they cover the cases that
will recur throughout the rest of the site.
Example: Sections in three settings
Let \(M\) be a smooth manifold with or without boundary.
(a) Tangent bundle. A section of \(TM\) is, by definition, a continuous map
\(X : M \to TM\) with \(X_p \in T_pM\) for each \(p\); this is precisely a
vector field
on \(M\). Smooth sections of \(TM\) are smooth vector fields. The development of vector fields
carried out earlier is therefore the development of (smooth) sections of \(TM\); the present
chapter generalizes it.
(b) Ambient tangent bundle. Given an immersed submanifold \(S \subseteq M\), a
section of the
ambient tangent bundle
\(TM|_S\) is a continuous map \(X : S \to TM\) with \(X_p \in T_pM\) for each \(p \in S\). Such
an \(X\) is called a
vector field along \(S\);
the value at \(p\) is a tangent vector to the ambient \(M\), not generally tangent to \(S\). The
gap between vector fields along \(S\) and vector fields on \(S\) — sections of \(TS\) instead
of \(TM|_S\) — encodes how \(S\) sits inside \(M\), and its analysis (via the splitting
\(TM|_S \cong TS \oplus NS\) when ambient Riemannian structure is available) is taken up later
in the development of submanifold geometry.
(c) Trivial bundle. Suppose \(E = M \times \mathbb{R}^k\) is the trivial
rank-\(k\) bundle. A continuous map \(F : M \to \mathbb{R}^k\) determines a section
\(\widetilde F : M \to M \times \mathbb{R}^k\) by \(\widetilde F(p) = (p, F(p))\), and every
section arises this way: the first coordinate is forced by \(\pi \circ \widetilde F = \mathrm{Id}_M\),
and the second coordinate is the function \(F\). The map \(F \leftrightarrow \widetilde F\) is a
bijection between continuous functions \(M \to \mathbb{R}^k\) and sections of the trivial bundle.
When \(M\) is smooth, \(\widetilde F\) is smooth if and only if \(F\) is.
(d) Trivial line bundle. The special case \(k = 1\) of (c) gives a natural
identification between the space \(C^\infty(M)\) of smooth real-valued functions on \(M\) and
the space of smooth sections of the trivial line bundle \(M \times \mathbb{R} \to M\). Smooth
functions on \(M\) are, in this sense, the simplest smooth sections that exist.
The trivial-bundle correspondence in (c)–(d) is the cleanest possible structural fact about sections:
on a globally trivial bundle, sections are nothing more than functions into the typical fiber. On a
non-trivial bundle this correspondence holds only locally, and the obstruction to globalizing it is
precisely the obstruction to triviality — a theme that recurs in the next section's discussion of
frames and trivializations.
The Algebra of Sections
Sections of a vector bundle can be added and scaled pointwise, using the vector space structure of
each fiber. They can also be multiplied by smooth real-valued functions on the base, with the result
being another section. These operations endow the set of smooth global sections with the structure
of a module over the ring of smooth functions on the base — the algebraic substrate on which the
later theory of tensor fields and differential operators is built.
Pointwise vector space structure
For sections \(\sigma_1, \sigma_2\) of a smooth vector bundle \(\pi : E \to M\) and real scalars
\(c_1, c_2 \in \mathbb{R}\), the pointwise combination
\[
(c_1 \sigma_1 + c_2 \sigma_2)(p) = c_1 \sigma_1(p) + c_2 \sigma_2(p)
\]
is well-defined because both \(\sigma_i(p)\) lie in the same fiber \(E_p\), which is itself a vector
space. Continuity (resp. smoothness) of \(c_1 \sigma_1 + c_2 \sigma_2\) follows because in any local
trivialization the operation reduces to pointwise linear combination of \(\mathbb{R}^k\)-valued
functions, which preserves continuity and smoothness.
Definition: The Space of Smooth Sections
The set of all smooth global sections of a smooth vector bundle \(\pi : E \to M\), equipped with
the pointwise vector space operations above, is denoted \(\Gamma(E)\). For the tangent bundle,
the alternative notation \(\mathfrak{X}(M)\) introduced for the space of smooth
vector fields
is the same object: \(\Gamma(TM) = \mathfrak{X}(M)\).
Multiplication by smooth functions
Beyond addition and scalar multiplication, smooth sections admit a richer operation: multiplication
by smooth functions on the base. The fiberwise action
\[
(f \sigma)(p) = f(p) \, \sigma(p)
\]
is well-defined since \(f(p) \in \mathbb{R}\) and \(\sigma(p) \in E_p\), and the product lies in
\(E_p\) by the vector space structure of the fiber. The result is again a smooth section: in a local
trivialization \(\Phi\), the composite \(\Phi \circ (f \sigma)\) sends \(p \in U\) to
\((p, f(p) w(p))\) where \(\Phi \circ \sigma(p) = (p, w(p))\) and \(w : U \to \mathbb{R}^k\) is
smooth; pointwise multiplication of \(w\) by \(f\) is smooth, so \(\Phi \circ (f \sigma)\) is
smooth, hence so is \(f \sigma\).
Proof:
Pointwise addition of smooth sections and multiplication of a smooth section by a smooth function
each produce a smooth section, as verified above. The module axioms hold because they hold
pointwise in each fiber \(E_p\), which is a vector space, and the operations on \(\Gamma(E)\) are
defined pointwise: for example, \((f + g)\sigma\) and \(f\sigma + g\sigma\) are sections whose
value at every \(p\) is \((f(p) + g(p))\sigma(p) = f(p)\sigma(p) + g(p)\sigma(p)\), and the two
therefore coincide as sections. The other axioms reduce to fiberwise vector space identities in
the same way.
The case \(E = TM\) recovers a structure already in hand: \(\mathfrak{X}(M)\) is a
\(C^\infty(M)\)-module,
a statement that until now was specific to vector fields and is recognized here as a special case of
a uniform fact about all smooth vector bundles. The pattern is general: every structural property of
\(\mathfrak{X}(M)\) that can be phrased in terms of the bundle structure of \(TM\) — without using
anything specific to tangent vectors — extends automatically to \(\Gamma(E)\) for arbitrary smooth
vector bundles.
What the module structure buys
Riemannian metrics, almost complex structures, symplectic forms, and tensor fields — the
tensorial geometric structures developed in later pages of the site — are most naturally defined
as \(C^\infty(M)\)-multilinear maps between \(\Gamma\)-modules of various bundles. A Riemannian
metric on \(M\), for instance, is a \(C^\infty(M)\)-bilinear map
\(g : \Gamma(TM) \times \Gamma(TM) \to C^\infty(M)\) satisfying positivity and symmetry; an
\((r,s)\)-tensor field is a multilinear map of the appropriate signature. (Connections, in
contrast, are not tensorial in this sense: they satisfy a Leibniz rule that fails
\(C^\infty(M)\)-linearity in one of their arguments, and live in a richer affine-structure
framework taken up separately.) The module structure established here is what makes these
tensorial definitions algebraically meaningful, replacing the ad-hoc fibrewise descriptions one
would otherwise be forced to use.
The Extension Lemma for Sections
A common technical need in differential geometry is to extend a section defined on a closed subset
to a global section, controlling where the extension is non-zero. The mechanism is the same one
that drove the
extension lemma for smooth functions:
smooth bump functions supplied by a
partition of unity
let local extensions be glued together. The statement for sections is parallel.
Lemma: Extension Lemma for Vector Bundles
Let \(\pi : E \to M\) be a smooth vector bundle over a smooth manifold \(M\) with or without
boundary, let \(A \subseteq M\) be a closed subset, and suppose \(\sigma : A \to E\) is a section
of \(E|_A\) that is smooth in the sense that it extends to a smooth local section of \(E\) in a
neighbourhood of each point of \(A\). For every open subset \(U \subseteq M\) containing \(A\),
there exists a global smooth section \(\widetilde \sigma \in \Gamma(E)\) such that
\(\widetilde \sigma|_A = \sigma\) and \(\mathrm{supp}\,\widetilde \sigma \subseteq U\).
Proof:
By the hypothesis on smoothness of \(\sigma\), for each \(p \in A\) there exist an open
neighbourhood \(W_p \subseteq M\) of \(p\) and a smooth local section
\(\sigma_p : W_p \to E\) with \(\sigma_p|_{W_p \cap A} = \sigma|_{W_p \cap A}\). By shrinking
\(W_p\) if necessary we may assume \(W_p \subseteq U\). The collection
\(\{W_p : p \in A\} \cup \{M \setminus A\}\) is an open cover of \(M\): every point of \(A\) lies
in some \(W_p\), and every point of \(M \setminus A\) lies in the open set \(M \setminus A\)
itself (the complement of a closed set).
Choose a smooth partition of unity \(\{\psi_p\}_{p \in A} \cup \{\psi_0\}\) subordinate to this
cover, with \(\mathrm{supp}\,\psi_p \subseteq W_p\) and \(\mathrm{supp}\,\psi_0 \subseteq M \setminus A\).
Each product \(\psi_p \sigma_p\), defined on \(W_p\), extends by zero to a smooth section on all
of \(M\): outside \(\mathrm{supp}\,\psi_p\) the section is zero, and on the open set \(W_p\) it
is smooth, so the extension is smooth on \(M\) (smoothness being a local condition, and the two
descriptions agreeing on the overlap). Define
\[
\widetilde \sigma = \sum_{p \in A} \psi_p \sigma_p ,
\]
where the sum is locally finite by the partition of unity property.
The section \(\widetilde \sigma\) is smooth as a locally finite sum of smooth sections. On \(A\),
each \(\sigma_p\) agrees with \(\sigma\) wherever both are defined, so
\(\psi_p \sigma_p|_{A} = \psi_p|_{A} \cdot \sigma\); summing and using
\(\sum_p \psi_p|_{A} = 1\) (because \(\psi_0|_A = 0\), as \(\mathrm{supp}\,\psi_0\) misses \(A\))
gives \(\widetilde \sigma|_A = \sigma\). For the support condition, each
\(\mathrm{supp}\,(\psi_p \sigma_p) \subseteq \mathrm{supp}\,\psi_p \subseteq W_p \subseteq U\),
and a locally finite union of closed sets contained in \(U\) is itself closed and contained in
\(U\), so \(\mathrm{supp}\,\widetilde \sigma \subseteq U\).
The case \(E = TM\) recovers the
extension lemma for vector fields
proved earlier in the manifold series. As with the module structure, what was previously a
TM-specific result is now visible as one instance of a bundle-wide pattern. The corollary one usually
extracts — that the value of a smooth section at a single point can be prescribed by choosing any
element of the fiber — confirms that \(E\) is fully populated by smooth global sections.
Corollary: Every Element of \(E\) Lies on a Smooth Global Section
Let \(\pi : E \to M\) be a smooth vector bundle. For every \(v \in E\), there exists a smooth
global section \(\sigma \in \Gamma(E)\) with \(\sigma(\pi(v)) = v\).
Proof:
Let \(p = \pi(v) \in M\). The singleton \(A = \{p\}\) is closed, and the assignment
\(p \mapsto v\) is a section of \(E|_A\); to verify smoothness in the sense of the extension
lemma, choose a smooth local trivialization \(\Phi : \pi^{-1}(W) \to W \times \mathbb{R}^k\)
around \(p\), write \(\Phi(v) = (p, w)\) for some \(w \in \mathbb{R}^k\), and define
\(\sigma_W(q) = \Phi^{-1}(q, w)\) for \(q \in W\). Then \(\sigma_W : W \to E\) is a smooth local
section with \(\sigma_W(p) = v\), exhibiting the local-extension property. The extension lemma
(with \(U = M\)) produces a global smooth section \(\widetilde \sigma \in \Gamma(E)\) with
\(\widetilde\sigma(p) = v\).
Local and Global Frames
The
local-frame
concept introduced for tangent bundles extends without change to general vector bundles. A frame is a
fiberwise-basis chosen smoothly across an open set, and it functions as a non-canonical "coordinate
system" for sections — every section is written uniquely as a smooth combination of frame elements.
The decisive structural fact, taken up in this section, is that smooth local frames and smooth local
trivializations are the same data viewed from two sides: each determines the other, and the
obstruction to one globalizing is the obstruction to the other.
Definition: Local and Global Frames for a Vector Bundle
Let \(\pi : E \to M\) be a vector bundle of rank \(k\), and let \(U \subseteq M\) be an open
subset. A \(k\)-tuple of local sections \((\sigma_1, \dots, \sigma_k)\) of \(E\) over \(U\) is
said to be linearly independent if for each \(p \in U\) the values
\((\sigma_1(p), \dots, \sigma_k(p))\) form a linearly independent \(k\)-tuple in \(E_p\), and
said to span \(E\) if their values span \(E_p\) for each \(p \in U\). A
local frame for \(E\) over \(U\) is an ordered \(k\)-tuple
\((\sigma_1, \dots, \sigma_k)\) of linearly independent local sections over \(U\) that span \(E\);
equivalently, \((\sigma_1(p), \dots, \sigma_k(p))\) is a basis of \(E_p\) at every \(p \in U\).
A local frame is called a global frame if \(U = M\). If \(E\) is a smooth vector
bundle, a local or global frame is called a smooth frame if each \(\sigma_i\) is
a smooth section. Following the convention adopted for tangent bundles, a frame
\((\sigma_1, \dots, \sigma_k)\) is often denoted \((\sigma_i)\).
For the tangent bundle, this definition agrees with the
local frame
defined earlier for \(TM\): a smooth local frame for \(TM\) over \(U\) is a \(k\)-tuple of smooth
vector fields whose values span every tangent space \(T_pM\) for \(p \in U\). The terms "frame for
\(M\)" and "frame for \(TM\)" mean the same thing and are used interchangeably.
Definition: The Frame Associated with a Local Trivialization
Let \(\pi : E \to M\) be a smooth vector bundle and
\(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) a smooth local trivialization. Let
\((e_1, \dots, e_k)\) denote the standard basis of \(\mathbb{R}^k\). The
local frame associated with \(\Phi\) is the \(k\)-tuple of smooth local sections
\((\sigma_1, \dots, \sigma_k)\) on \(U\) defined by
\[
\sigma_i(p) = \Phi^{-1}(p, e_i) , \qquad p \in U .
\]
Each \(\sigma_i\) is smooth as the composition of the smooth map \(p \mapsto (p, e_i)\) with the
diffeomorphism \(\Phi^{-1}\), and the relation \(\pi \circ \Phi^{-1} = \pi_U\) on its image
ensures \(\pi(\sigma_i(p)) = p\), so \(\sigma_i\) is a section. Linear independence and spanning
at each \(p\) follow because \(\Phi\) restricts to a vector space isomorphism
\(E_p \to \{p\} \times \mathbb{R}^k\) sending \((\sigma_i(p))\) to \((e_i)\).
For the product bundle \(E = M \times \mathbb{R}^k\) the identity trivialization yields the global
frame \(\widetilde e_i(p) = (p, e_i)\). For a general bundle, applying the construction over each
trivializing open set produces a covering of \(M\) by smooth local frames, none of which need extend
to a global frame.
Completion of local frames
Independent partial frames can always be completed to full frames locally. The next proposition
records three versions of this completion — from a partial frame on an open set, from a linearly
independent tuple at a single point, or from a frame defined on a closed subset.
Proposition: Completion of Local Frames for Vector Bundles
Let \(\pi : E \to M\) be a smooth vector bundle of rank \(k\).
-
If \((\sigma_1, \dots, \sigma_m)\) is a linearly independent \(m\)-tuple of smooth local
sections of \(E\) over an open subset \(U \subseteq M\), with \(1 \le m < k\), then for each
\(p \in U\) there exist smooth sections \(\sigma_{m+1}, \dots, \sigma_k\) defined on some
neighbourhood \(V\) of \(p\) such that \((\sigma_1, \dots, \sigma_k)\) is a smooth local
frame for \(E\) over \(U \cap V\).
-
If \((v_1, \dots, v_m)\) is a linearly independent \(m\)-tuple of elements of \(E_p\) for some
\(p \in M\), with \(1 \le m \le k\), then there exists a smooth local frame
\((\sigma_i)\) for \(E\) over some neighbourhood of \(p\) such that \(\sigma_i(p) = v_i\) for
\(i = 1, \dots, m\).
-
If \(A \subseteq M\) is a closed subset and \((\tau_1, \dots, \tau_k)\) is a linearly
independent \(k\)-tuple of sections of \(E|_A\) that are smooth in the sense described in the
extension lemma, then there exists a smooth local frame \((\sigma_1, \dots, \sigma_k)\) for
\(E\) over some neighbourhood of \(A\) such that \(\sigma_i|_A = \tau_i\) for \(i = 1, \dots, k\).
Proof:
We prove the three parts in order.
(a) Fix \(p \in U\) and choose a smooth local trivialization
\(\Phi : \pi^{-1}(V_0) \to V_0 \times \mathbb{R}^k\) on a neighbourhood \(V_0\) of \(p\). For
\(i = 1, \dots, m\) write \(\Phi(\sigma_i(q)) = (q, w_i(q))\) for \(q \in U \cap V_0\), where
\(w_i : U \cap V_0 \to \mathbb{R}^k\) is smooth. The \(m\)-tuple \((w_1(p), \dots, w_m(p))\) is
linearly independent in \(\mathbb{R}^k\) by hypothesis, so it can be completed to a basis
\((w_1(p), \dots, w_m(p), w_{m+1}, \dots, w_k)\) by choosing constant vectors
\(w_{m+1}, \dots, w_k \in \mathbb{R}^k\). Linear independence is an open condition in matrix
space (it is the complement of the zero set of the determinant of the \(k \times k\) matrix
\([w_1(q), \dots, w_m(q), w_{m+1}, \dots, w_k]\)), so there is a neighbourhood
\(V \subseteq V_0\) of \(p\) on which the tuple
\((w_1(q), \dots, w_m(q), w_{m+1}, \dots, w_k)\) remains linearly independent for all
\(q \in V\). Define \(\sigma_j(q) = \Phi^{-1}(q, w_j)\) for \(j = m+1, \dots, k\) and
\(q \in V\); each \(\sigma_j\) is smooth (composition of a smooth map with a diffeomorphism),
and at each \(q \in U \cap V\) the values \((\sigma_1(q), \dots, \sigma_k(q))\) are the images
under the fiberwise isomorphism \(\Phi^{-1}\) of a basis of \(\{q\} \times \mathbb{R}^k\), hence
a basis of \(E_q\).
(b) Choose a smooth local trivialization \(\Phi\) on a neighbourhood \(V_0\) of
\(p\) and write \(\Phi(v_i) = (p, w_i)\) for \(i = 1, \dots, m\). Linear independence of
\((v_1, \dots, v_m)\) in \(E_p\) is equivalent to that of \((w_1, \dots, w_m)\) in
\(\mathbb{R}^k\), so by part (a) applied to the constant local sections
\(q \mapsto \Phi^{-1}(q, w_i)\) (for \(i = 1, \dots, m\)), one obtains smooth completing
sections \(\sigma_{m+1}, \dots, \sigma_k\) on a neighbourhood \(V\) of \(p\). Setting
\(\sigma_i(q) = \Phi^{-1}(q, w_i)\) for \(i = 1, \dots, m\) gives a smooth local frame
\((\sigma_1, \dots, \sigma_k)\) on \(V\) with \(\sigma_i(p) = v_i\) for \(i = 1, \dots, m\).
(c) By the extension lemma applied to each \(\tau_i\) (with \(U\) any open set
containing \(A\)), there exist smooth global sections
\(\widetilde\tau_1, \dots, \widetilde\tau_k \in \Gamma(E)\) restricting to
\(\tau_1, \dots, \tau_k\) on \(A\). At each \(p \in A\), the values
\((\widetilde\tau_1(p), \dots, \widetilde\tau_k(p)) = (\tau_1(p), \dots, \tau_k(p))\) form a
basis of \(E_p\). Linear independence is an open condition, so the set
\(W = \{q \in M : (\widetilde\tau_i(q))_{i=1}^k \text{ is linearly independent in } E_q\}\) is an
open neighbourhood of \(A\), and \((\widetilde\tau_1|_W, \dots, \widetilde\tau_k|_W)\) is a
smooth local frame on \(W\) extending \((\tau_i)\) on \(A\).
Frame ↔ trivialization duality
The construction above — local trivialization producing local frame — has an inverse: every smooth
local frame arises from a smooth local trivialization. This is the structural fact that organizes
the entire theory of sections.
Proposition: Every Smooth Local Frame Comes from a Local Trivialization
Let \(\pi : E \to M\) be a smooth vector bundle of rank \(k\), and let \((\sigma_i)\) be a smooth
local frame for \(E\) over an open subset \(U \subseteq M\). There exists a unique smooth local
trivialization \(\Psi : \pi^{-1}(U) \to U \times \mathbb{R}^k\) such that the local frame
associated with \(\Psi\) is \((\sigma_i)\) in the sense of the preceding definition.
Proof:
We use the frame data to write down a candidate for \(\Psi^{-1}\), then verify that it is a
diffeomorphism by comparing it with an existing local trivialization. Define
\(\Psi : U \times \mathbb{R}^k \to \pi^{-1}(U)\) by
\[
\Psi\bigl(p, (v^1, \dots, v^k)\bigr) = v^i \sigma_i(p) .
\]
We use the convention that \(\Psi\) here is the inverse trivialization; the trivialization
itself, written \(\Phi_\sigma = \Psi^{-1}\), is what is described in the statement.
Bijectivity. For each \(p \in U\), the values \((\sigma_1(p), \dots, \sigma_k(p))\)
form a basis of \(E_p\) by the frame hypothesis. Thus the map
\((v^1, \dots, v^k) \mapsto v^i \sigma_i(p)\) is the basis-coordinate isomorphism
\(\mathbb{R}^k \to E_p\), bijective for each \(p\). It follows that \(\Psi\) is a bijection from
\(U \times \mathbb{R}^k\) to \(\pi^{-1}(U)\), with \(\pi \circ \Psi = \pi_U\). The composite
\(\sigma_i = \Psi \circ \widetilde e_i\) (where \(\widetilde e_i(p) = (p, e_i)\)) recovers the
sections from \(\Psi\), so the local frame associated with \(\Phi_\sigma = \Psi^{-1}\) is
\((\sigma_i)\), as required.
Smoothness. Both \(\Psi\) and \(\Psi^{-1}\) must be shown to be smooth. We show
this by relating them to a known local trivialization. Fix \(q \in U\). By the local-triviality
of \(E\), there exists a smooth local trivialization
\(\Phi : \pi^{-1}(V) \to V \times \mathbb{R}^k\) on some open neighbourhood \(V\) of \(q\); by
shrinking \(V\) we may assume \(V \subseteq U\). It suffices to show that \(\Psi|_{V \times \mathbb{R}^k}\)
is a diffeomorphism onto \(\pi^{-1}(V)\).
Each \(\sigma_i|_V\) is a smooth section, so the composite
\(\Phi \circ \sigma_i : V \to V \times \mathbb{R}^k\) is smooth, with image of the form
\((p, \sigma_i^*(p))\) for a smooth map \(\sigma_i^* : V \to \mathbb{R}^k\). Write the
components of \(\sigma_i^*\) as \(\sigma_i^j : V \to \mathbb{R}\), so
\[
\Phi \circ \sigma_i(p) = \bigl(p,\, (\sigma_i^1(p), \dots, \sigma_i^k(p))\bigr) , \qquad p \in V .
\]
Composing \(\Phi\) with \(\Psi\) on \(V \times \mathbb{R}^k\) gives
\[
\Phi \circ \Psi\bigl(p, (v^1, \dots, v^k)\bigr)
= \Phi(v^i \sigma_i(p))
= \bigl(p,\, v^i \sigma_i^*(p)\bigr)
= \bigl(p,\, (v^i \sigma_i^1(p), \dots, v^i \sigma_i^k(p))\bigr) ,
\]
which is smooth in \((p, v)\).
For the inverse direction, note that at each \(p \in V\) the matrix
\(A(p) = [\sigma_i^j(p)]_{j,i}\) is the change-of-basis matrix from \((\sigma_i(p))\) to the
basis of \(E_p\) determined by \(\Phi\), so \(A(p) \in GL(k, \mathbb{R})\). Let
\(\tau(p) = A(p)^{-1}\); because matrix inversion is a smooth map
\(GL(k, \mathbb{R}) \to GL(k, \mathbb{R})\), the map \(p \mapsto \tau(p)\) is smooth. The inverse
\(\Psi^{-1} \circ \Phi^{-1}\) on \(V \times \mathbb{R}^k\) is then
\[
\Psi^{-1} \circ \Phi^{-1}\bigl(p, (w^1, \dots, w^k)\bigr)
= \bigl(p,\, (w^j \tau_j^1(p), \dots, w^j \tau_j^k(p))\bigr) ,
\]
smooth in \((p, w)\). Both directions being smooth, the restriction of \(\Psi\) to
\(V \times \mathbb{R}^k\) is a diffeomorphism. Since every point of \(U\) lies in such a
\(V\), \(\Psi\) is a diffeomorphism on \(U \times \mathbb{R}^k\), and \(\Phi_\sigma = \Psi^{-1}\)
is the required smooth local trivialization.
Uniqueness: any smooth local trivialization \(\Phi'\) whose associated frame is \((\sigma_i)\)
must send \(\sigma_i(p)\) to \((p, e_i)\) for each \(p, i\), and by linearity on fibers must
therefore agree with \(\Phi_\sigma\) on every fiber. Thus \(\Phi' = \Phi_\sigma\).
Two corollaries follow at once. The first identifies smooth triviality with the existence of a
smooth global frame; the second packages a smooth local frame and a smooth chart on the base into a
smooth chart on the total space.
Corollary: Smooth Triviality and Global Frames
A smooth vector bundle is smoothly trivial if and only if it admits a smooth global frame.
Proof:
Apply the frame ↔ trivialization correspondence above with \(U = M\): smooth global frames and
smooth global trivializations correspond bijectively. A smoothly trivial bundle is one admitting
a smooth global trivialization, equivalently a smooth global frame.
Specialized to the tangent bundle, this is the statement that \(TM\) is smoothly trivial if and only
if \(M\) is
parallelizable;
spheres of dimensions \(1\), \(3\), and \(7\) are parallelizable (and \(\mathbb{S}^2\) is not), but
the general classification is delicate and lies outside our scope. For Lie groups, however, the
answer is clean and is part of the
Lie algebra construction:
every Lie group is parallelizable, with the left-invariant vector fields supplying a smooth global
frame for \(TG\).
Corollary: Natural Smooth Chart on E from Frame and Base Chart
Let \(\pi : E \to M\) be a smooth vector bundle of rank \(k\), let \((V, \varphi)\) be a smooth
chart on \(M\) with coordinate functions \((x^i)\), and suppose there exists a smooth local
frame \((\sigma_i)\) for \(E\) over \(V\). Define
\(\widetilde \varphi : \pi^{-1}(V) \to \varphi(V) \times \mathbb{R}^k\) by
\[
\widetilde\varphi\bigl(v^i \sigma_i(p)\bigr) = \bigl(x^1(p), \dots, x^n(p),\, v^1, \dots, v^k\bigr) .
\]
Then \((\pi^{-1}(V), \widetilde \varphi)\) is a smooth coordinate chart for \(E\).
Proof:
Let \(\Phi_\sigma\) be the smooth local trivialization associated with the frame \((\sigma_i)\)
on \(V\) (the preceding proposition). Then \(\widetilde \varphi\) factors as the composition
\((\varphi \times \mathrm{Id}_{\mathbb{R}^k}) \circ \Phi_\sigma\): the first map sends a fiber
element to its trivialization coordinates, and the second sends the base point to the chart
coordinates while preserving fiber components. Both factors are diffeomorphisms, so
\(\widetilde\varphi\) is a diffeomorphism onto its image, hence a smooth coordinate chart.
Smoothness via component functions
Let \((\sigma_i)\) be a smooth local frame for \(E\) over \(U \subseteq M\), and let
\(\tau : M \to E\) be a rough section. At each \(p \in U\) the value \(\tau(p)\) lies in
\(E_p\), and the basis \((\sigma_1(p), \dots, \sigma_k(p))\) determines unique scalars
\((\tau^1(p), \dots, \tau^k(p))\) such that \(\tau(p) = \tau^i(p) \sigma_i(p)\). The functions
\(\tau^i : U \to \mathbb{R}\) so defined are called the component functions of \(\tau\)
with respect to the local frame \((\sigma_i)\). Smoothness of the section translates into
smoothness of the component functions, and conversely.
Proposition: Local Frame Criterion for Smoothness
Let \(\pi : E \to M\) be a smooth vector bundle, let \(\tau : M \to E\) be a rough section, and
let \((\sigma_i)\) be a smooth local frame for \(E\) over an open subset \(U \subseteq M\). Then
\(\tau\) is smooth on \(U\) if and only if its component functions
\((\tau^i)\) with respect to \((\sigma_i)\) are smooth.
Proof:
Let \(\Phi_\sigma : \pi^{-1}(U) \to U \times \mathbb{R}^k\) be the smooth local trivialization
associated with the frame \((\sigma_i)\). Because \(\Phi_\sigma\) is a diffeomorphism, \(\tau\)
is smooth on \(U\) if and only if \(\Phi_\sigma \circ \tau : U \to U \times \mathbb{R}^k\) is
smooth. By the definition of the associated trivialization, \(\Phi_\sigma(\sigma_i(p)) = (p, e_i)\),
so by linearity on fibers
\[
\Phi_\sigma(\tau(p)) = \Phi_\sigma\bigl(\tau^i(p) \sigma_i(p)\bigr) = \bigl(p,\, (\tau^1(p), \dots, \tau^k(p))\bigr) .
\]
Thus \(\Phi_\sigma \circ \tau\) is smooth on \(U\) if and only if the component functions
\(\tau^i\) are smooth on \(U\).
This proposition applies equally well to local sections defined over an open subset \(V \subseteq M\)
(such a local section being a global section of the restricted bundle \(E|_V\)). Combined with the
frame ↔ trivialization duality, it says that smoothness of sections, smoothness of component
functions, and existence of trivializations are three views of the same structure — the choice
among them being pedagogical, not mathematical.
What sections, frames, and trivializations are
Three objects have been linked in this section. A section assigns to each base point a
vector in its fiber. A frame over an open set assigns to each point a basis of its
fiber, varying smoothly with the point. A trivialization over an open set identifies
the bundle restricted to that set with the product of the set and a fixed model fiber. The
correspondences are: a frame determines a trivialization (Proposition above), a trivialization
determines a frame (its associated frame from the definition above), and sections in either
viewpoint are smooth functions into the model fiber. Globally, the question "is the bundle
trivial?" is the same as "does a global frame exist?" is the same as "is there an isomorphism
with the product bundle?". The Möbius bundle fails all three; the tangent bundles of spheres
\(\mathbb{S}^n\) fail them for all \(n\) other than \(1, 3, 7\); Lie group tangent bundles
satisfy them universally.
The Smooth Structure on TM Revisited
The tangent bundle of a smooth manifold was constructed earlier by assembling tangent spaces and
promoting the resulting set to a smooth manifold via natural coordinate charts. At the time, the
smooth structure looked like one of several plausible choices: a particular chart system was singled
out, and other choices might in principle have produced different bundles. The vocabulary now in
hand — sections, frames, trivializations — lets us prove that this is not the case. The smooth
structure on \(TM\) is the only one compatible with the geometric content already present: that
coordinate vector fields are smooth and that \(TM\) is a smooth vector bundle.
The uniqueness statement
Proposition: Uniqueness of the Smooth Structure on TM
Let \(M\) be a smooth \(n\)-manifold with or without boundary. The topology and smooth structure
on \(TM\) constructed in the
smooth structure on the tangent bundle
are the unique ones with respect to which \(\pi : TM \to M\) is a smooth vector bundle, with the
given vector space structure on the fibers, and such that all coordinate vector fields
\(\partial/\partial x^i\) (associated with smooth charts on \(M\)) are smooth local sections.
Proof:
Suppose \(TM\) is endowed with some topology and smooth structure making it a smooth vector
bundle with the stated properties; we show this structure is equal to the one constructed
previously.
Let \((U, \varphi)\) be any smooth chart on \(M\) with coordinate functions \((x^i)\). By
hypothesis the
coordinate vector fields
\((\partial/\partial x^1, \dots, \partial/\partial x^n)\) are smooth local sections of \(TM\)
over \(U\), and they form a basis of each tangent space \(T_pM\) for \(p \in U\); they are
therefore a smooth local frame for \(TM\) over \(U\) with respect to the given smooth structure.
Applying the frame ↔ trivialization correspondence
(Proposition above) to this smooth local frame yields a smooth local trivialization
\(\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^n\) with respect to the given smooth structure on
\(TM\), characterized by the property that
\[
\Phi\!\left(v^i \frac{\partial}{\partial x^i}\bigg|_p\right) = \bigl(p, (v^1, \dots, v^n)\bigr) .
\]
This is, however, the very map constructed in the proof that
TM is a smooth vector bundle
— exhibiting that the smooth structure built on \(TM\) earlier in the manifold series is one
with respect to which the same \(\Phi\) is a local trivialization. By the corollary on natural
smooth charts (Corollary above), the composite
\[
\widetilde \varphi = (\varphi \times \mathrm{Id}_{\mathbb{R}^n}) \circ \Phi
: \pi^{-1}(U) \to \varphi(U) \times \mathbb{R}^n
\]
is a smooth chart for \(TM\) with respect to the given smooth structure. But \(\widetilde\varphi\)
is precisely the natural coordinate chart on \(TM\) used in the
original construction
to define its smooth structure. Thus every chart of the original smooth structure belongs to
the given one. By the same argument applied with the roles swapped, every chart of the given
smooth structure belongs to the original one. The two maximal smooth atlases coincide, so the
smooth structures are equal.
The proposition closes a circular dependency that has been quietly present since the tangent bundle
was first built. The smooth structure on \(TM\) was used to define smooth vector fields, which
were used to define smooth local frames, which were used here to recover the smooth structure on
\(TM\). The circle is virtuous: the apparently auxiliary choice of charts on the total space turns
out to be determined by the algebraic data — what sections are smooth, what frames are available
— that the bundle structure organizes. The smooth structure and the section algebra encode the
same information.
Three further observations close the section. The first concerns the role of the coordinate-vector-field
hypothesis. The hypothesis is not optional: the bare requirement that \(TM\) be a smooth vector
bundle, without specifying which sections must be smooth, admits other smooth structures
(corresponding to different choices of admissible local frames). The coordinate vector fields are
the geometric data that pin down the structure, and the
component transformation law
is what makes them coherent across overlapping charts. Second, the uniqueness extends to every
"natural" bundle built from \(TM\) — the cotangent bundle, the tensor bundles, the exterior power
bundles — by a parallel argument that will appear in later development, with the coordinate basis
of each fiber playing the role that \((\partial/\partial x^i)\) plays here. Third, although the
uniqueness statement targets \(TM\) specifically, the proof relies only on the general frame ↔
trivialization correspondence; the same structure of argument characterizes any smooth vector
bundle whose smooth structure is constrained by a distinguished local frame on each coordinate
chart of the base.
The closing arc
The manifold series began with the question of when a topological space admits a unique smooth
structure compatible with given local data (the smooth manifold chart lemma). It ends with the
analogous question for vector bundles, answered first in general (the vector bundle chart lemma
on the previous page) and then specialized to the most important case (the tangent bundle,
this page). What started as construction has become characterization. For the tangent bundle,
once one demands that the bundle structure interact correctly with the coordinate vector
fields, the smooth structure is forced. This same pattern — local algebraic-or-geometric data
determining the smooth structure uniquely — recurs in later developments: cotangent bundles,
tensor bundles, and exterior power bundles inherit unique smooth structures from \(TM\) by
analogous arguments, and a Riemannian metric on \(M\) is the additional data that lets one
construct further intrinsic structures on these bundles (connections, curvature, geodesics)
beyond what the bundle framework alone provides.