Let's consider the unit sphere $S^2 \subset \mathbb{R}^3$, $U_0=S^2-\{(0,0,-1)\}$ and $U_1=S^2-\{(0,0,1)\}$ and spherical coordinates $(\phi,\theta)$. For $n\ \in \mathbb{Z}$ we can consider $\omega_0=\frac{-in}{2}\frac{sin^2(\phi)}{1+cos(\phi)}d\theta \in \Omega^1(U_0)$ and $\omega_1=\frac{in}{2}\frac{sin^2(\phi)}{1-cos(\phi)}d\theta\in \Omega^1(U_1)$. I have already checked that they define local connection forms for a complex line bundle $L_n\rightarrow S^2$ with transition functin $g_{01}:(\phi,\theta)\rightarrow e^{in\theta}$.
Now if we consider $C_{\phi}$ to be the north spherical cap bounded by the parallel $\gamma_{\phi}(\theta)=(\cos(\theta)\sin(\phi),\sin(\theta)\sin(\phi),\cos(\phi))$, $\theta \in[0,2\pi]$ , $C_{\phi}=\{(x,y,z)\in S^2: z>\cos(\phi)\}$. I want to see that $\gamma_{\phi}^* L_n$ has a parallel section if and only if $\int_{C_{\phi}}\Omega \in 2\pi i \mathbb{Z}$, where $\Omega$ is the curvature form.
I am bit confused about something since I thought that we would always have that a paralllel section exists since we can solve an ordinary differential equation to get the parallel tranport. In this specific case in local coordinates with respect to $U_0$ we get that $s(t)=v(t)s_0(c(t))$, where $s_0$ is a linear independent section on $U_0$, and so we get that we wanna solve the differential equation $\nabla_{\frac{d}{dt}}s(t)=\frac{d}{dt}v(t)+(\frac{-in}{2}\frac{\sin^2(\phi)}{1+\cos(\phi)})v(t)=0$. And I guess a solution for this is just going to be $v_0\exp(\frac{in}{2}\frac{\sin^2(\phi)}{1+\cos(\phi)})$.
After analyzing this for a while we could get that condition on the curvature if we ask that $s(0)=s(2\pi)$, but I am not sure why one needs to do this.
Any help is appreciated.
Thanks in advance.