Line bundles associated to principal circle bundles Let $\pi: P \rightarrow B$ be a principal circle bundle over $B$ and $\rho: S^1 \times \mathbb{C} \rightarrow \mathbb{C}$ an effective left action. Then, one can associate to the bundle $\pi$ a complex line bundle $\pi_{\rho}:P \times_{\rho} \mathbb{C} \rightarrow B$ by the canonical projection, where $$ P \times_{\rho} \mathbb{C} := \{[p,z]\in P \times \mathbb{C}\,|\, (p,z) \equiv (p\cdot\theta, \rho(\theta,z))\  \text{for some}\  \theta \in S^1 \} . $$
My question is the following: Define two left circle actions 
$\rho_{1}, \rho_{2}: S^1 \times \mathbb{C} \rightarrow \mathbb{C}$ by
$$\rho_{1}(\theta, z)=e^{i\theta}z, \ \ \rho_{2}(\theta,z)=e^{-i\theta}z.$$
Then,
1) Are two associated bundle $\pi_{\rho_{j}}:P \times_{\rho_{j}}\mathbb{C} \rightarrow B$ ($j=1,2$) isomorphic as vector bundles?; 
2) Are the two total spaces $P \times_{\rho_{j}}\mathbb{C}$ ($j=1,2$) mutually diffeomorphic? 
I am happy to get to know the answer to each question. Thank you in advance.
 A: Some time has passed, so I'm not sure if you still need the answer, but here it is:
The two line bundles will be duals of each other, so they will not be isomorphic, unless it's the trivial bundle.
Suppose $s_i$ are local sections of the circle bundle given in the following way. If $\phi_i : \pi^{-1}(U_i) \rightarrow U_i \times S^1$ is the local trivialisation of the bundle, then its inverse can be expressed as $\phi^{-1}_i(x,\theta) \rightarrow \theta + s_i(x)$ for some local section $s_i$ (I'm using the additive notation here so that exponentiation makes sense as a homomorphism later on). If $\psi_{ji}$ are transtion funtions for the bundle (going form $i$ to $j$ trivialisation), then the realtionship between $s$s is $s_i(x) = \psi_{ji}(x) + s_j(x)$.
Now we trivialize the corresponding complex lines in the following way 
$$\pi_\rho^{-1}(U_i) \ni [s_i(x),z] \rightarrow (x,z) \in U_i\times \mathbb{C}.$$
Then it is not hard to show that the transition functions for this line bundles are $e^{i\psi_{ji}}$ and $e^{-i\psi_{ji}}$ for the actions $\rho_1$ and $\rho_2$ respectively. That also shows that the trivialisation was well-defined since equivalent circle bundles will produce eqiuvalent line bundles. Finally, note that line bundles are duals of each other when their transition functions are inverses of each other.
