Hermann Weyl wrote his classic account of “The Idea of a Riemann Surface” in 1913, and it quickly became a classic. In 1955, he produced a revised edition, preserving some of his original approach but taking more careful account of the developments since then. The revised edition was translated as The Concept of a Riemann Surface and published by Addison-Wesley in 1955. This is the edition Dover has now brought back into print.
Weyl writes in the introduction that
When German mathematicians and the publishing house of Teubner approached me with the invitation to prepare a new edition, since requests for the book continued, it at first seemed appropriate to treat the book more or less as an historical document and send it into the world again unchanged except for a few minor improvements. But as I attempted to merge the appendix with the main text, I became ever more conscious of the deficiencies of both the appendix and the text.
In other words, those looking for information on the history of topology, and especially on the development of the notion of a manifold, will need to look at both versions of the book. In fact, comparing the 1913 version with this one should yield interesting insights.
One of the main changes Weyl made in the book consists in moving from a more combinatorial approach (triangulations, for example) to an approach along the lines of the modern theory of differentiable manifolds. As he points out, the resulting theory is very close to Chevalley’s approach in his Introduction to the Theory of Algebraic Functions of One Variable (AMS, 1951, and still in print). Weyl was also influenced by the work of Kodaira and Hodge on complex manifolds of higher dimension, which clearly served as a source of inspiration.
The resulting book is a valuable short introduction to the theory of complex functions in one variable, dense and rewarding. In 1913, it stood alone. Today, it will share shelf space with many other texts on the subject, but its personal approach and depth of insight still make it worth our time.
“I have followed my own ideas too much,” says Weyl, “and have paid too little attention to other ideas, especially those in the potent literature of topology. May I not be judged too harshly!” Far from it, Herr Professor.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He is the editor of MAA Reviews.