Mathematics / Fiber Bundles / Vector bundles and local frames

Fiber Bundles notes

Chern Classes and Bloch Bundles Fiber Bundles, Connections, and Curvature Principal Bundles and Connections Topological Fiber Bundles Vector Bundles, Frames, and Sections

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Vector Bundles From Scratch

The problem vector bundles solve

In ordinary linear algebra, we work with one fixed vector space $V$ . But on a manifold, the natural vector space may depend on the point. For example, at each point $p\in M$ there is a tangent space $T_pM$ . These spaces are all isomorphic to $\mathbb{R}^n$ , but there is usually no canonical way to identify $T_pM$ with $T_qM$ for two different points $p,q\in M$ .

A vector bundle is a controlled way of saying:

There is a vector space attached to every point of the base space, and these vector spaces vary continuously or smoothly from point to point.

Formal definition

Definition 12 (Real vector bundle). A rank- $r$ real vector bundle over a topological space $B$ is a topological fiber bundle

\pi:E\to B

with fiber $\mathbb{R}^r$ , such that:

Each fiber $E_b=\pi^{-1}(b)$ is an $r$ -dimensional real vector space.
There exists an open cover $\{U_i\}$ of $B$ and local trivializations

\phi_i:\pi^{-1}(U_i)\to U_i\times\mathbb{R}^r

that restrict on each fiber $E_b$ to a linear isomorphism

E_b\to \{b\}\times\mathbb{R}^r\cong\mathbb{R}^r.

On overlaps, transition functions take values in $\mathop{\mathrm{GL}}(r,\mathbb{R})$ .

A complex vector bundle is the same definition with $\mathbb{C}^r$ and $\mathop{\mathrm{GL}}(r,\mathbb{C})$ .

If $B=M$ is a smooth manifold and $E$ is also a smooth manifold, then a smooth vector bundle requires smooth local trivializations and smooth transition functions

g_{ij}:U_i\cap U_j\to \mathop{\mathrm{GL}}(r,\mathbb{R})

g_{ij}:U_i\cap U_j\to \mathop{\mathrm{GL}}(r,\mathbb{C}).

Definition 13 (Line bundle). A rank- $1$ vector bundle is called a line bundle. A rank- $1$ real vector bundle is a real line bundle, with fiber $\mathbb{R}$ . A rank- $1$ complex vector bundle is a complex line bundle, with fiber $\mathbb{C}$ .

For a real line bundle, the transition functions take values in

\mathop{\mathrm{GL}}(1,\mathbb{R})=\mathbb{R}^\times,

the nonzero real numbers under multiplication. If a fiber metric is chosen and local frames are normalized, the structure group reduces to

O(1)=\{+1,-1\}.

This is why the Möbius strip can be described using transition functions equal to $+1$ or $-1$ .

Is $E$ arbitrary or equal to $M\times\mathbb{R}^{\dim M}$ ? The total space $E$ is not assumed to be $M\times\mathbb{R}^{\dim M}$ . It is an independent space equipped with a projection $\pi:E\to M$ and local product structures. Only the trivial rank- $r$ bundle is globally $M\times\mathbb{R}^r$ . The rank $r$ is independent of $\dim M$ except in special examples such as the tangent bundle, where $r=\dim M$ .

If $M$ is an $n$ -dimensional smooth manifold and $E\to M$ is a rank- $r$ real smooth vector bundle, then the total space $E$ is locally modeled on

U\times\mathbb{R}^r\subset \mathbb{R}^n\times\mathbb{R}^r,

so $E$ is an $(n+r)$ -dimensional smooth manifold. For the tangent bundle $TM\to M$ , $r=n$ , so $TM$ has dimension $2n$ .

The trivial bundle

The simplest vector bundle is the product

E=M\times\mathbb{R}^r.

The projection is

\pi(p,v)=p.

The fiber over $p$ is

E_p=\{p\}\times\mathbb{R}^r\cong\mathbb{R}^r.

This is called the trivial rank- $r$ bundle.

A bundle is trivial if there exists a global trivialization

E\cong M\times\mathbb{R}^r

compatible with the projection and vector-space structures. Many bundles are locally trivial but not globally trivial.

Local frames

Let $E\to M$ be a rank- $r$ vector bundle. On an open set $U\subset M$ , a local frame is a collection of $r$ local sections

e_1,\ldots,e_r

such that for every $p\in U$ , the vectors

e_1(p),\ldots,e_r(p)

form a basis of $E_p$ .

Choosing a local frame is equivalent to choosing a local trivialization. If a section $s$ is defined on $U$ , then

s(p)=e_a(p)\psi^a(p).

The functions $\psi^a$ are the local components of $s$ .

Transition functions for vector bundles

Suppose we have two patches $U_i$ and $U_j$ with local frames

e^{(i)}_1,\ldots,e^{(i)}_r, \qquad e^{(j)}_1,\ldots,e^{(j)}_r.

On the overlap $U_i\cap U_j$ , both frames are valid. They must be related by an invertible matrix:

e_a^{(j)}=e_b^{(i)}\,g_{ij}^{ba}.

In matrix notation,

e^{(j)}=e^{(i)}g_{ij}.

The map

g_{ij}:U_i\cap U_j\to \mathop{\mathrm{GL}}(r,\mathbb{R})

is called a transition function. For a complex vector bundle,

g_{ij}:U_i\cap U_j\to \mathop{\mathrm{GL}}(r,\mathbb{C}).

If the bundle has a Hermitian metric and we choose orthonormal frames, then

g_{ij}:U_i\cap U_j\to U(r).

On triple overlaps,

g_{ij}g_{jk}=g_{ik}.

This is the vector-bundle cocycle condition.

Analogy with projective representations In projective representations, associativity forces a cocycle condition on phase factors:

U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2).

For bundles, consistency of patch gluing forces

g_{ij}g_{jk}=g_{ik}.

These are not the same mathematical object, but they express a common idea: local data must agree consistently on triple overlaps or triple products.

Sections

Let $\pi:E\to M$ be any vector bundle. A section is a smooth map

s:M\to E

such that

\pi\circ s=\mathrm{id}_M.

This means $s(p)\in E_p$ for every $p\in M$ .

Notation:

\Gamma(E)=\text{space of smooth sections of }E.

This notation is used constantly in differential geometry. For example,

\Gamma(TM)=\text{vector fields on }M,

\Gamma(T^*M)=\text{one-forms on }M,

\Omega^k(M)=\Gamma(\Lambda^kT^*M)=\text{$k$-forms on }M.

If $E=M\times\mathbb{R}^r$ is trivial, a section is the same thing as a smooth function

\psi:M\to\mathbb{R}^r,

because it has the form

s(p)=(p,\psi(p)).

For a nontrivial vector bundle, sections are still locally vector-valued functions, but globally they need not be expressible as maps $M\to\mathbb{R}^r$ .

How section components transform

Let

s=e^{(i)}\psi_i=e^{(j)}\psi_j.

Since $e^{(j)}=e^{(i)}g_{ij}$ ,

e^{(i)}\psi_i=e^{(i)}g_{ij}\psi_j.

Therefore

\psi_i=g_{ij}\psi_j.

Physics translation The section $s$ is the global geometric object. The column vector $\psi_i$ is only its coordinate description in patch $U_i$ . A gauge transformation changes the description, not the underlying section.

Gauge changes as changes of frame

Suppose on each patch $U_i$ we change frame by

e^{(i)}\mapsto \widetilde e^{(i)}=e^{(i)}h_i,

where

h_i:U_i\to \mathop{\mathrm{GL}}(r,\mathbb{R})

or $\mathop{\mathrm{GL}}(r,\mathbb{C})$ .

The transition functions change by

g_{ij}\mapsto \widetilde g_{ij}=h_i^{-1}g_{ij}h_j.

Thus transition functions are not individually gauge-invariant. The bundle is the equivalence class of such gluing data under changes of local frames.

Canonical Vector Bundles on a Manifold

The tangent bundle

For each point $p\in M$ , there is a tangent space $T_pM$ . Collect all tangent spaces:

TM:=\bigsqcup_{p\in M}T_pM.

The symbol $\bigsqcup$ means disjoint union: even if two tangent vectors have the same coordinate components, they are regarded as different if they live at different base points.

The projection is

\pi:TM\to M, \qquad \pi(v_p)=p.

This is the tangent bundle.

Important sentence The tangent bundle $TM\to M$ is a rank- $n$ vector bundle over an $n$ -dimensional manifold $M$ .

A vector field is a smooth choice of one tangent vector at each point:

X(p)\in T_pM.

Therefore a vector field is a map

X:M\to TM

satisfying

\pi\circ X=\mathrm{id}_M.

Such a map is called a section.

Vector field = section A vector field is not the tangent bundle itself. It is a section of the tangent bundle.

\text{vector field }X \in \Gamma(TM).

The cotangent bundle

Similarly, collect the cotangent spaces:

T^*M:=\bigsqcup_{p\in M}T_p^*M.

This is the cotangent bundle.

A one-form is a smooth choice of one covector at each point:

\alpha(p)\in T_p^*M.

So a one-form is a section of the cotangent bundle:

\alpha\in\Gamma(T^*M).

More generally, $k$ -forms are sections of the bundle

\Lambda^kT^*M\to M.

Precise language $TM$ , $T^*M$ , and $\Lambda^kT^*M$ are vector bundles over $M$ . Vector fields and differential forms are sections of these bundles, not the bundles themselves.

First Examples of Nontrivial Bundles

Real line bundles

A real line bundle over a base space $B$ is a rank- $1$ real vector bundle

\pi:L\to B.

This means that every fiber

L_b=\pi^{-1}(b)

is a one-dimensional real vector space, hence isomorphic to $\mathbb{R}$ . Locally,

\pi^{-1}(U_i)\cong U_i\times\mathbb{R}.

On overlaps, transition functions are nonzero real-valued functions

g_{ij}:U_i\cap U_j\to \mathbb{R}^\times.

If we choose a fiber metric and normalized local frames, then only signs remain:

g_{ij}:U_i\cap U_j\to O(1)=\{+1,-1\}.

The trivial real line bundle over $B$ is

B\times\mathbb{R}\to B.

A nontrivial real line bundle is locally $U\times\mathbb{R}$ but cannot be globally written as $B\times\mathbb{R}$ in a way compatible with the vector-space structure on fibers.

The Möbius line bundle

The Möbius strip is the simplest nontrivial real line bundle over $S^1$ .

At every point of $S^1$ , attach a copy of $\mathbb{R}$ . Locally it looks like

U\times\mathbb{R}.

But after going once around the circle, the fiber coordinate flips sign:

v\mapsto -v.

The structure group is

O(1)=\{+1,-1\}.

The nontrivial transition function is the sign $-1$ .

Moral of the Möbius example Locally, the Möbius bundle is indistinguishable from the cylinder $S^1\times\mathbb{R}$ . Globally, it is twisted. This is the basic phenomenon fiber bundles are designed to capture.

A real line bundle is trivial if it admits a nowhere-zero global section. The Möbius line bundle does not: any attempted nonzero section comes back with opposite sign after going around the circle.

Complex line bundles

A complex line bundle has fiber $\mathbb{C}$ . Its structure group can be taken to be

\mathop{\mathrm{GL}}(1,\mathbb{C})=\mathbb{C}^\times.

With a Hermitian metric, it reduces to

U(1).

Thus transition functions are maps

g_{ij}:U_i\cap U_j\to U(1).

A complex line bundle is the natural home for wavefunctions whose phase can only be chosen locally.