Let $E\to M$ be a smooth $\mathbb{K} = \mathbb{R}, \mathbb{C}$ - vector bundle over a possibly non-compact connected manifold $M$. Denote by $\mathbb{P}(E) \to M$ its projectivization, which is obtained by removing the zero section of $E$ and fiberwise taking the projective quotient $\sim$ which identifies lines on each fiber:

$\mathbb{P}(E) = \frac{E\backslash 0_M}{\sim}$

Denote by $\pi\colon E\backslash 0\to \mathbb{P}(E)$ the canonical projection, which pointwise sends an element of a fiber to the class it defines in projective space. As I understand, $\mathbb{P}(E)$ may not admit any section. Assume it does admit a smooth section $s\in \mathbb{P}(E)$. I am interested in the obstruction to lift $s\colon M\to \mathbb{P}(E)$ to a nowhere vanishing section $\eta\colon M\to E$ of $E$ such that $\pi(\eta) = s$. A quick computation in $\check{\mathrm{C}}$ech cohomology shows that, given $s$, there is a unique obstruction $c(s)\in H^1(M,\mathbb{K}^{\ast})$ for lifting $s$ to a section of $E$ that projects to $E$. Now, I am not sure if this is a characteristic class of $E$, or if it depends on the section $s$ chosen (whose existence may be obstructed but I assume). I have googled the literature but I have not found this problem discussed anywhere. Notice that this is different from the problem (extensively discussed in the literature) of finding the obstruction for a projective bundle to be the projectivization of a vector bundle. Here that is taken for granted and the obstruction corresponds to lifting a section.

Thanks.