The Riemann Sphere

This chapter begins with a discussion of stereographic projection and its properties, that circles on the sphere are mapped to generalised circles on the plane and that stereographic projection is conformal. This leads to the Riemann Sphere, viewed as a Riemann surface, and a characterisation of meromorphic functions on the Riemann sphere. The spherical metric on the extended complex plane is constructed as a Riemannian metric and its isometries characterised. The chapter concludes with a brief discussion of spherical geometry.