First I'll briefly outline the physical context in which my question arose: If one tries to do 2D quantum mechanics in polar coordinates one encounters a problem with a property of the covariant derivative $\nabla$. Namely that the the radial momentum operator $P_r$ defined in terms of it $P_r:=-i\nabla_r=-i\partial_r$ (if the object to be derivated is a function $f:\mathbb{R^2}→ \mathbb{C}$) is no longer self adjoint. According to this lecture https://www.youtube.com/watch?v=C93KzJ7-Es4&t=1630s one can address this issue by interpreting the wave function $\Psi$ (i.e. the object to be derivated) no longer as a function $f_\Psi:\mathbb{R^2}→ \mathbb{C}$ but as a section $\Psi:\mathbb{R^2}→E$ on a $\mathbb{C}$-line bundle $E$ over $\mathbb{R^2}$. It is claimed that $E$ is also an associated fibre bundle to the frame bundle $L\mathbb{R^2}$ over $\mathbb{R^2}$ which can be equipped with a connection $\omega$. And for such $\Psi \in \Gamma(E)$ the covariant derivative contains an expression with $\omega$ which can be adjustet to get a $P_r=-i\nabla_r$ which is again self adjoint.
Now my problem with the argumentation: The frame bundle $L\mathbb{R^2}$ is trivial as one can easily find a global section of it. Every associated vector bundle of a trivial principal bundle is trivial too. But if $E$ is trivial a section $\Psi$ can be understood as a function $f_\Psi:\mathbb{R^2}→\mathbb{C}$, which is the starting point. The covariant derivative of such an object is of course again $\nabla_r=\partial_r$. My question now is: What is wrong in my derivation of a $\nabla_r$ for the section $\Psi$? Because I don't think my formula is valid and we just choose the one where a term with $\omega$ arises.