Do two associated bundles (one the associated of the other) share same transition functions in general? (If yes why? If no why?)
Can anyone give me a reference where I can find the rigorous definition?
Thanks.
2026-03-25 20:39:39.1774471179
Associated bundles
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You get the associated bundle trans. functions just composing with the action you are considering. E.g. if the $G$-principal bundle has trans. fun $g_{\beta \alpha}:U_{\alpha}\cap U_\beta \to G$ and the action you are considering is $\rho:G\to \mathrm{Aut}(V)$ then the transition functions of the assiciated bundle are given by $\rho\circ g_{\beta \alpha}$.
The classic reference would be Kobayashi-Nomizu vol 1 but they don't play much with transition functions. You can find this perspective more used in Nicolaescu "Lectures on the Geometry of Manifolds".