ABSTRACT
This chapter introduces the basic concepts and theorems in Riemann surface theory. Because the local coordinate transisitions of a Riemann surface is bi-holomorphic, the angles are well defined on the whole surface. The concepts of holomorphic and meromorphic functions can be generalized to Riemann surfaces. A memromorphic function can be treated as a holomorphic map from the Riemann surface to the unit sphere. Here we only prove the dimension of the linear space of holomorphic 1-forms. A surface with a Riemannian metric is called a Riemannian surface. A Riemannian metric structure is stronger than a conformal structure, because the metric can measure both the angles and the areas, but the conformal structure can only measure the angles. Furthermore, a Riemannian metric can induce a compatible conformal structure.
