Associated vector bundle

A vector bundle obtained from a principal bundle and a linear representation of its structure group.

Associated vector bundle

Associated vector bundles are a special case of associated bundles where the fiber is a vector space and the structure group acts linearly.

Let $\pi:P\to M$ be a principal G-bundle with right action $(p,g)\mapsto p\cdot g$ . Let $\rho:G\to GL(V)$ be a representation of a Lie group . View this as a left action of $G$ on $V$ via

g\cdot v := \rho(g)\,v.

(Compare the convention in associated bundles use a left G-action on the fiber .)

The associated vector bundle with fiber $V$ is

E := P\times_G V := (P\times V)/\sim,

where the equivalence relation is

(p\cdot g,\; v)\sim (p,\; g\cdot v)\qquad\text{for all }p\in P,\ g\in G,\ v\in V.

Write the equivalence class of $(p,v)$ as $[p,v]$ .

The projection map $\pi_E:E\to M$ is defined by

\pi_E([p,v]) := \pi(p).

With this projection and the linear structure on the fibers inherited from $V$ , $E$ becomes a vector bundle over $M$ .

This construction is the specialization of the associated bundle construction P times_G F to the case $F=V$ .

Local trivializations and transition functions

Given a local section $s:U\to P$ (see local trivialization from a local section ), there is a canonical vector bundle trivialization

\Phi_s: E|_U \longrightarrow U\times V,\qquad \Phi_s([s(x),v])=(x,v).

If $s_i$ and $s_j$ are local sections on $U_i$ and $U_j$ with $s_i(x)=s_j(x)\,g_{ij}(x)$ on overlaps, then the corresponding transition function for $E$ is

(x,v)\longmapsto \bigl(x,\ \rho(g_{ij}(x))\,v\bigr),

so the vector bundle transition functions are obtained by composing the principal bundle transition functions (see transition function ) with the representation $\rho$ .

Examples

Tangent bundle as an associated bundle Let $M$ be an $n$ -manifold and let $Fr(M)\to M$ be its frame bundle , a principal $GL(n,\mathbb{R})$ -bundle. Using the standard representation of $GL(n,\mathbb{R})$ on $\mathbb{R}^n$ , the associated vector bundle is canonically isomorphic to the tangent bundle:
$TM \cong Fr(M)\times_{GL(n,\mathbb{R})}\mathbb{R}^n.$
Complex line bundles from principal $U(1)$ -bundles If $P\to M$ is a principal $U(1)$ -bundle and $\rho:U(1)\to GL(1,\mathbb{C})$ is the standard character $\rho(e^{i\theta})=e^{i\theta}$ , then
$L := P\times_{U(1)}\mathbb{C}$
is a complex line bundle. For instance, using the Hopf fibration $S^3\to S^2$ as $P$ , this produces the tautological line bundle over $S^2\cong \mathbb{CP}^1$ .
Adjoint (Lie algebra) bundle Taking $V=\mathfrak{g}$ with the adjoint representation gives the associated bundle
$\mathrm{Ad}(P)=P\times_G \mathfrak{g},$
which is the usual adjoint Lie algebra bundle (see construction of the adjoint Lie algebra bundle ).