in my phd thesis, informaly I use this result:
Let $(A_i,\omega_i)$, $i=1,2$, where $A_i$ is an annulus and $\omega_i$ is an abelian differential form with a translation structure on $A_i$ that continuously extends to $\gamma_i$, where $\gamma_i$ is a geodesic of $\omega_i$. If $$\displaystyle\int_{\gamma_1}\omega_1=\displaystyle\int_{\gamma_2}\omega_2$$
Then we can glue $(A_1,\omega_1)$ with $(A_2,\omega_2)$ by identifying $\gamma_1\sim\gamma_2$ to obtain a translation structure in $\dfrac{A_1\displaystyle\cup A_2}{\gamma_1\sim\gamma_2}$. The gluing is done by fixing a point on $\gamma_1$ and varying the points on $\gamma_2$. Thus, there are as many ways to paste as there are points on $\gamma_2$.
I write "Informally" because it is intuitive, I would like you to help me to formalize it, I have not found any book about this. A geodesic is the curve that minimize the distance between two points, but I do not understand the geodesic, $\gamma$, induced by an abelian form, $\omega$ and their relation with the integral. I will really appreciate any hint. Thank you!