Fiber Bundle

Remainder

Lie group

  • Definition (Lie group $G$)
    1. is a manifold
    2. is a (continuos) group
    3. has the group operations:

    $$ \begin{align*} \bullet &: G\times G \to G \\ & (g_1, g_2) \to g_1 \cdot g_2 \\ ^{-1} &: G \to G \\ & g \to g^{-1} \end{align*} $$

  • Examples: $GL(n, \mathbb{R})$, $SL(n, \mathbb{R})$, $SL(n, \mathbb{C})$ ,$SO(n)$, $SU(n)$, $U(n)$

Action of Lie group

  • Definition (Left Action of Lie group $G$ on a manifold $\mathcal{M}$) is $\sigma : G\times\mathcal{M} \to \mathcal{M}$ where $\sigma$ a smooth map such that
    1. $\sigma(e, p) = p \forall p\in \mathcal{M}$
    2. $\sigma(g_1, \sigma(g_2, p)) = \sigma(g_{1}.g_{2}, p)$. This means we have an operation $g\cdot p$ where $e\cdot p = p$ and $g_{1}(g_{2}\cdot p) = (g_{1}g_{2})\cdot p$. So, $g_1 g_2$ is left action.
  • Analogously, one can define a right action.
  • Example: $SO(3)$ on $\mathbb{R}^{3}$ (or $S^{2}$) acts by standard rotation. GL(n, \mathbb{R})) on $\mathbb{R}^{n}$ has left action $\sigma({ A^{i}_{\hspace{0.1cm}j} }, { x^{k} }) = (A^{i} _{\hspace{0.1cm}j} x^{j}) \in \mathbb{R}^{n}$

Definition

  • A (differential) fibre bundle $(\mathcal{E}, \pi, \mathcal{M}, \mathcal{F}, \mathcal{G})$ where

    1. A manifold $\mathcal{E} \equiv$ “Total space”
    2. A manifold $\mathcal{M} \equiv$ “Base space”
    3. A manifold $\mathcal{E} \equiv$ “Fibre” (or typical fibre)
    4. A surjection $\pi : \mathcal{E} \to \mathcal{M}$ called the projection $\forall_{p\in \mathcal{M}} \pi_{-1} (p) = F_{p} \cong F$ i.e. Fibre at point $p$ is diffeomorphic to fibre. We use $\circ$ notation to refer the map $\pi$.
    5. $G$ is a Lie group which acts on $F$ on the left. We call $F$ as structure group.
    6. An open covering ${ U_{i} }$ of $\mathcal{M}$ with diffeomorphism $\phi_{i} : U_{i}\times F \mapsto \pi^{-1}(U_{i})$ such that $(\pi \circ \phi_{i}) (p, F) = p$ where $\phi_{i}$ is called local trivialization as $\phi_{i}^{-1}$ maps $\pi^{-1} (U_{i})$ onto the cartesian product $U_{i}\times F$. If we fix point $p$, we get a diffeomorphism:

    $$ \begin{align*} \phi_{i, p} &: F \to F_{p} \\ \phi_{i, p}(F) &= \phi_{i}(p, F). \end{align*} $$

    1. For $U_{i} \cap U_{j} \neq \phi$, the maps $\phi_{i}$ and $\phi_{j}$ are related by a fiber-wise left action of the structure group. i.e.

    $$ \begin{align*} \forall p \in U_{i} \cap U_{j}, \phi_{i, p} &: F \mapsto F_{p} \\ \phi_{j, p} &: F \to F_{p} \end{align*} $$ then $$ \begin{align*} \phi_{i, p}^{-1} \circ \phi_{j, p} : F \mapsto F \end{align*} $$ is given by left action by $t_{ij}(p) \in G$ so that $$ \begin{align*} \phi_{j}(p, f) = \phi_{i}(p, t_{ij}(p)f) \end{align*} $$ where $t_{ij} : U_{i} \cap U_{j} \mapsto G$ is a transition function (a smooth map).

Examples

  1. The tangent bundle is an explicit example

    $$ \begin{align*} \mathcal{E} &= \mathcal{T}\mathcal{M}\quad \text{dim}\mathcal{M} = m \\ F &= \mathbb{R}^{m} \\ G &= GL(m, \mathbb{R}) \\ \end{align*} $$ Given coordinate patch $U_{i}$ with co-ordinate $(X^{\mu}, V^{\mu})$ and $U_{j}$ with $\left(X^{\mu'}, \frac{\partial X^{\mu'}}{\partial X^{\mu}} V^{\mu} \right)$ then, $(t_{ij})^{\mu'}_{\hspace{0.1cm} \mu} \in GL(m, \mathbb{R})$.

Properties of transition function

  • $t_{ii}(p) = \text{id}$ for $p \in U_{i}$
  • $t_{ij}(p) = t_{ji}(p)^{-1}$ for $p \in U_{i}\cap U_{j}$
  • $t_{ij}(p)\cdot t_{jk}(p) = t_{ik}(p)$ for $p \in U_{i}\cap U_{j} \cap U_{k}$.

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Published on Aug 8, 2021

Last revised on Apr 3, 2022

References