$G$-bundles

An alternative approach to fibre bundles is that one of the $G$-bundles:

Definition

Let $M$, $S$ be manifolds. We say $E\stackrel{\pi}{\rightarrow}M$ is a $G$-bundle over $M$ if we can find an open covering $\{U_{\alpha}\}$ for $M$, together with a collection of homeomorphisms (local trivializations)

$$ \phi_{\alpha}: U_{\alpha}\times S\longrightarrow \pi^{-1}(U_{\alpha}) $$

such that

For $m\in U_{\alpha}$, we have a map

$$ \tilde{\phi}_{\alpha}(m)(s):=\phi_{\alpha}(m,s) $$

for $s\in S$ such that is a diffeomorphism from $S$ to $\pi^{-1}(\{m\})$.

If $U_{\alpha\beta}:=U_{\alpha}\cap U_{\beta}\neq 0$ the transition functions

$$ g_{\alpha \beta}(m):=\tilde{\phi}_{\alpha}^{-1}(m)\circ \tilde{\phi}_{\beta}(m) $$

satisfy $g_{\alpha\beta}(m)\in G \subseteq \textrm{Diff}(S)$.

$\blacksquare$

We remark the following observations:

For $m\in M$, $E_m:=\pi^{-1}(\{m\})$ is called the fibre of $E$ over $m$. Since $S$ is diffeomorphic to every fibre $E_m$, is usually called the standard fibre.

The collection $\{(U_{\alpha},\phi_{\alpha})\}$ is called a trivializing atlas for $E\rightarrow M$.

We have considered $G$ as a subgroup of $\textrm{Diff}(S)$, but often is more convenient to think of $G$ as an abstract group with a faithful left action $\lambda$ over $S$. We often denote the action by

$$ \lambda(g_{\alpha\beta}(m),s) =g_{\alpha\beta}(m)\cdot s $$

or even by

$$ \lambda(g_{\alpha\beta}(m),s) =g_{\alpha\beta}(m) (s) $$

when it does not result in confusion.

For an open set $U\subseteq M$, any map

$$ \sigma: U \longrightarrow E $$

such that $\pi \circ \sigma=id_M$ is called a local section of the fibre bundle $E\stackrel{\pi}{\rightarrow} M$. A local section $\sigma$ defined on a trivializing open set $U_{\alpha}$ can be identified with a map

$$ \tilde{\sigma}_{\alpha}:U_{\alpha}\longrightarrow S $$

such that $\sigma(x)=\phi_{\alpha}(x,\tilde{\sigma}_{\alpha}(x))$.

The best example for a $G$-bundle is a vector bundle, where $S$ is a vector space $V$ and we take $G=\textrm{GL}(V)$.

Another important case: principal bundles.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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