7-2-68. Felix Klein to H. Poincaré, English translation
Leipzig, 14 May 1882
In response to the letter that just arrived, I would like to explain briefly how I employ “continuity”. Only in principle, of course, since the detailed exposition – which will be quite troublesome to write up – can in any case be qualified in many ways. I will confine myself throughout to the case of unbranched -functions of the second kind, as I called them in my note. Here it is primarily a question of proving that the two manifolds to be compared – on the one hand, the set of systems of substitutions considered, and on the other, the set of all existing Riemann surfaces – not only have the same dimension ( real dimensions) but form analytic manifolds with analytic boundaries (in the sense of the terminology introduced by Weierstrass). By the first lemma stated in my previous letter, these two manifolds are related in a -valued manner, where by the second lemma, , for the various parts of the second manifold, can take on only the values 0 or 1. Now this relation turns out to be analytic and, as this again follows from both lemmas, an analytic relation whose functional determinant vanishes nowhere. From this I deduce that always has the value 1. Let there be a transition from a region with to another with , then, in view of the analyticity of the correspondence, the first region would correspond to definite points (those actually attained) of the other region, and then for these points the functional determinant of the relation would have to vanish, contrary to the supposition. So much for my proof.
Mr. Schwarz put forward a quite different proof, also based on considerations of continuity, when I visited him recently (on April 11) in Göttingen. Although I haven’t obtained his permission to do so, I thought I should tell you about it all the same. He imagines the Riemann surface cut appropriately and then covered infinitely often with the various sheets so connected along the cuts that a complete surface results corresponding to a set of polygons placed side by side in the plane. This complete surface, if one may so term an infinitely extended surface (which still requires clarification), is, in the case of an -function of the second kind (the case first considered by Schwarz), simply connected and with simple boundary curve, and then it is a question of seeing whether such a simply connected surface with simple boundary can be mapped in the usual way onto the interior of a disc. Schwarz’s train of thought is very beautiful, in any case.
You ask about offprints. I would, above all, not wish to trouble you about that, particularly since I can always procure of all your works with the sole exception of your Thèse. So if you can send me a few items (I don’t have any so far), they would be most welcome.
Have you perhaps had the chance to read Lie’s theory of transformation groups? Lie always takes the parameter figuring in his groups to be complex-valued; it would be interesting to see how his results extend to the situation where one considers groups generated only by a real iteration of certain small operations.
Some time ago Hermite sent me one of the lithographed lessons of his Cours d’analyse. Would it be possible (subject to payment of course) to obtain the whole set? This would be particularly well met, in light of the aims I’m currently pursuing in my seminar.
As always, your most devoted,
PTrL. Translated by R. Burns from the original German (§ 4-47-23) in Saint-Gervais (2016), with minor revisions by S.A. Walter. See also the French translation (§ 7-2-41), and Poincaré’s reply (§ 4-47-24).
Time-stamp: "14.05.2021 01:35"
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. Cited by: 7-2-68. Felix Klein to H. Poincaré, English translation.