I've been looking a lot at various texts on how to construct fiber bundles, but I'm having a bit of a challenge looking at the specific case of constructing a fiber bundle $\pi:E \to M$ where $M$ is the base space (a smooth manifold), $E$ is the total space (also a smooth manifold) and each fiber is diffeomorphic (or essentially is) a fixed homogeneous space $F \approx G/H$.
I know that a few ways to prove that we get a smooth fiber bundle include:
- Provide local trivializations $\Phi: U\times F \to \pi^{-1}(U)$, such that if $\Phi, \Psi$ are two such mappings whose images overlap, then the transition function $\Phi^{-1}\circ \Psi$ are smooth and the homogeneous coordinates are related by a group action.
- Possibly obtain a quotient of the Frame bundle over $M$ using the action $G$ of the group which defines our homogeneous space $F$.
- Use the disjoint union topology to define our total space to be $E = \bigsqcup_{p \in M} \{p\}\times F$ where we attach to each point $p\in M$ the desired fiber $F$.
While the last method listed above, using the disjoint union topology, seems like the most obvious way to do this, the homogeneous space I have in mind doesn't lend itself well to defining local trivializations since it's not clear to me how to relate the coordinate charts of $M$ to those of $F$. In fact, it's not even clear to me how to obtain coordinate charts for $F$ (though we can certainly pick arbitrary ones if that ends up being sufficient).
I know this question might come off as a little vague, but I'd appreciate any help that one can provide in helping me prove such a construction forms a fiber bundle.