Riemann’s Pupils and Followers

In selecting representative figures connected with the reception and development of Riemannian ideas, priority has been given here to those who knew Riemann personally. Among the Italians, Enrico Betti became a close personal friend and did much to promote Riemann's work, as did Felice Casorati. Probably the most important of Riemann's auditors in Göttingen were Karl Hattendorff, Friedrich Prym, and Gustav Roch. Riemann entrusted Hattendorff with writing up his paper on minimal surfaces, a task that the latter had to complete himself and then publish posthumously. Hattendorff also prepared two monographs based on Riemann's lectures, one of which dealt with partial differential equations in mathematical physics and went through four editions. Prym studied briefly under Riemann before going on to write his dissertation under Kummer in Berlin on a topic in Riemannian function theory. He spent a month with the Riemanns in Pisa and later took a professorship in Würzburg, where Adolf Krazer was one of his students. Gustav Roch was the most talented of Riemann's students, but he died just months after his mentor. His name is remembered today in connection with the Riemann-Roch equality for the space of meromorphic functions defined on a Riemann surface. Roch's notes on Riemann's lectures were published by Heinrich Weber in the Collected Works. Two other key figures for the reception of Riemann's work were Alfred Clebsch and his protégé Felix Klein. Neither was steeped in anlysis, so their achievements aimed either to apply or interpret Riemannian ideas. Clebsch had considerable success doing so in algebraic geometry. Klein initially developed a new way of visualizing the Riemann surfaces associated with real algebraic curves, but more influential was his booklet on Riemann's theory of algebraic functions and their integrals, which showed hos these can be studied by means of stationary currents on closed surfaces of a given genus. This presentation appealed greatly to Hermann Weyl, who referred to it in his well-known book Die Idee der Riemannschen Fläche.