TITLE: To differently gluing of two Riemann surfaces with boundary we get different surfaces
QUESTION [7 upvotes]: If $M,N$ are two Riemann surfaces with boundary, then we can glue them along one of each of their boundary component, which is $S$, to form a new Riemann surface with boundary, but for different gluing we may form different Riemann surfaces with boundary, for example, there may be a $S$ twist, intuitively, it is just we rotate one $S$ an angle then glue it with another surfaces, but my question is how can we show the resulting two surfaces (twisted gluing and untwisted gluing) are different Riemann surface with boundary (their differential structures are the same because we can regard it as a kind of connected sum)? Thanks!

REPLY [2 votes]: When you glue $M$ and $N$, for example, along some boundary component $C$. Choose a closed geodesic $\gamma$ on $S$ intersect with $C$. It can be shown that the hyperbolic length of $\gamma$ is changed when you twist along $C$. This means that different twist makes different hyperbolic surface.