Riemann surfaces
reception by the French school
pp. 237-291
in: Lizhen Ji, Athanase Papadopoulos, Sumio Yamada (eds), From Riemann to differential geometry and relativity, Berlin, Springer, 2017Abstract
Riemann introduced in his doctoral dissertation (1851) the concept of Riemann surface as a new ground space for meromorphic functions and in particular as a domain for a multi-valued function defined by an algebraic equation such that this function becomes single-valued when its is defined on its associated Riemann surface. It took several years to the mathematical community to understand the concept of Riemann surface and the related major results that Riemann proved, like the so-called Riemann existence theorem stating that on any Riemann surface—considered as a complex one-dimensional manifold—there exists a non-constant meromorphic function. In this chapter, we discuss how the concept of Riemann surface was apprehended by the French school of analysis and the way it was presented in the major French treatises on the theory of functions of a complex variable, in the few decades that followed Riemann's work. Several generations of outstanding French mathematicians were trained using these treatises. At the same time, this will allow us to talk about the remarkable French school that started with Cauchy and expanded in the second half of the nineteenth century. We also comment on the relations between the French and the German mathematicians during that period.