I'm reading the first chapter of the book "Geometry and spectra of compact Riemann surfaces" by P. Buser. On page 9, he wrote
Parametrize $\gamma$ and $\gamma^\prime$ with unit speed and opposite boundary orientation such that $\gamma(0) = a$ and $\gamma^\prime(0) = b$.
Why is "opposite boundary orientation"? I though that it should be the same orientation according to the picture he gave. Did I misunderstand something here?
Thank you!
it is the opposite orientation as part of the boundary. A consistent orientation would be, say, the path through $a,$ then a line segment, then the path through $b$ backwards of the drawn arrow, then another line segment. Riemann surfaces are orientable, the interior of the bounded region should be thought of as having a little arrow indicating a direction of rotation.