Let $M$ be a smooth manifold on which acts the $G$-action $\Phi$. According to the quotient manifold theorem, if $\Phi$ is free and proper, then the orbit space $M/G$ has a unique smooth manifold structure such that the projection map $\pi:M\mapsto M/G=\bar M:m\mapsto G.m$ is a smooth submersion and defines a principal fiber bundle with structure group $G$. A fiber of $\pi$ is a $G$-orbit in $M$ and is then diffeomorphic to the structure group $G$ with diffeomorphism $G.m\mapsto G:\Phi( g,m)\mapsto g$.

I would like to know if they are necessary or sufficient conditions on the action $\Phi$ that ensures that the principal fiber bundle is trivial. Thanks for any suggestions or references.


Principal bundles are trivial if and only if they admit a global smooth section $\sigma: M/G \to M$. Given such a section, consider the map $$f : G \times M/G \to M: (g, b) \mapsto \Phi(g, \sigma(b)).$$

The map $f$ is smooth because $\Phi$ and $\sigma$ are and it is a bijection because the action is free and transitive on the fibres. Since $\pi \circ f(g,b)=b$, identifying $M$ with $G \times M/G$ via $f$ shows that it is trivial.

Conversely, for a trivial bundle $G \times M/G$ we have obvious global sections $b \mapsto (g,b)$.


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