Let $\pi\colon P \to M$ be a principal bundle with structure group $G$ with respect to the Lie group right action $\triangleleft\colon P \times G \to P$.
To my knowledge, a reduction of the structure group of $\pi\colon P \to M$ to a Lie subgroup $\hat G \subset G$ is a subbundle $\hat \pi\colon \hat P \to M$ that is a principal bundle with respect to the restriction of $\triangleleft$ to $\hat P \times \hat G$.
For some reason I intuitionally suspected that the triviality of the principal bundle descends to its reductions of the structure group.
However, I now think that I found a counter-example and would like to confirm my suspicion:
Consider the torus $\mathbb T^2$ as total space of a trivial $U(1)$-principal bundle over $S^1$. We can define the Lie group action $\triangleleft\colon \mathbb T^2 \times U(1) \to \mathbb T^2$ to map a point $\begin{pmatrix}(1+r \sin(\psi)) \cos(\phi) \\ (1+r \sin(\psi)) \sin(\phi) \\ r \cos(\psi)\end{pmatrix}$ under the action of $e^{i \varphi}$ to a point given by $\begin{pmatrix}(1+r \sin(\psi+\varphi)) \cos(\phi) \\ (1+r \sin(\psi+\varphi)) \sin(\phi) \\ r \cos(\psi+\varphi)\end{pmatrix}$.
We then can choose a non-trivial reduction of the structure group to $\mathbb Z_2$ according to the red submanifold in the sketch below:
Questions
- Does my example work?
- If so, are there criteria for when the triviality of the principal bundle descends to its reductions of the structure group? (Other than the base space being contractible.)
- I was aware that there are non-trivial vector subbundles of trivial vector bundles. I however thought this question was unrelated to the case of principal bundles. Is it valid that the fact that there exist non-trivial vector subbundles of trivial vector bundles implies that there also exist non-trivial principal subbundles of trivial principal bundles? (Showing that the frame bundle of the non-trivial vector subbundle is a principal subbundle of the framebundle of the trivial vector bundle that we started with...)