7-2-66. Felix Klein to H. Poincaré, English translation

Leipzig, 7 May 1882

Sophienstraße 10

Dear Sir,

I recently read your note in the Comptes rendus of 10 April 1882.11endnote: 1 Poincaré (1882); reed. Nörlund and Lebon (1916, 41–43). The note was all the more interesting to me, as I believe your current considerations, and also your methods, to be closely related to mine. I prove my theorems using continuity, relying on the the following two lemmas: 1) to every “groupe discontinu” there always belongs a Riemann surface, and 2) to each suitably cut Riemann surface there always* belongs just one such group (if any group at all belongs to it).

Up to now I have not tried to derive the series developments that you establish. How in fact do you prove the existence of the number $m$ for which $\sum\frac{1}{(\gamma_{i}\eta+\delta_{i})^{m}}$ converges absolutely? And do you have an exact lower bound or an approximate one?

As for me, in the meantime I have been able to give the theorems in question an even more general form, but I shall have to write to you about that later, as I am pressed for time and need to prepare a note for the Annalen. In the case of my first theorem, the whole of the closed sphere $\eta$ with infinitely many points removed is covered by the images of the fundamental region. In the case of the second theorem, the interior of a disc, but of one only, remains uncovered. I have now noted the existence of representations (which for each Riemann surface are always unique) for which the case of infinitely many discs is excluded. In this direction, I formulate here only the the simplest theorem (in which it is assumed always that one has an unbranched representation of the Riemann surface).

Let $p=\mu_{1}+\mu_{2}+\ldots+\mu_{m}$, where none of the $\mu=1$. Choose $m$ points $O_{1}$, …, $O_{m}$ on the Riemann surface, and starting from $O_{1}$ make, in the usual way, $2\mu_{1}$ cuts $A_{1}$, $B_{1}$; $A_{2}$, $B_{2}$; …; $A_{\mu_{1}}$, $B_{\mu_{1}}$; from $O_{2}$ [make] $2\mu_{2}$ cuts, etc. On the other hand, on the sphere $\eta$, draw $m$ pairwise disjoint circles and in the interior of the space bounded by the latter taken together a polygon bounded by $4\mu_{1}$ circular arcs orthogonal to the first fundamental circle, then one bounded by $4\mu_{2}$ circular arcs orthogonal to the second fundamental circle, etc. (so that each polygon of circular arcs is $m$-fold connected). The bounding circles are ordered in pairs in accordance with the familiar sequence $A_{1}$, $B_{1}$, $A_{1}^{-1}$, $B_{1}^{-1}$, $A_{2}$, $B_{2}$, …, that is, by means of linear substitutions acting on $\eta$ in each case leaving the fundamental circle invariant. Furthermore, assume the product of the linear transformations in question – for example $A_{1}B_{1}A_{1}^{-1}B_{1}^{-1}\ldots A_{\mu_{1}}^{-1}B_{\mu_{1}}^{-1}$ – is always the identity. Then there is always one and only one analytic function mapping the sectioned Riemann surface onto one of the given polygons of circular arcs. The case when one of the $\mu$ is equal to 1 differs only in that then the corresponding fundamental circle reduces to a point and the corresponding linear substitutions become “parabolic”, and fix that point. So, enough for today. Would it be impossible to obtain a complete collection of your relevant offprints? After Pentecost I will be giving a series of lectures in my seminar on single-valued functions admitting linear transformations, and would, if possible, like to make available such a collection to my auditors.

Very respectfully yours,

F. Klein

* i.e., under the constraints of the theorem in question.

PTrL. Translated by R. Burns from the original German (§ 4-47-21) in Saint-Gervais (2016), with slight revisions by S.A. Walter. See also the French translation (§ 7-2-40).

Time-stamp: " 6.05.2021 19:56"

Notes

• 1 Poincaré (1882); reed. Nörlund and Lebon (1916, 41–43).

References

• N. E. Nörlund and E. Lebon (Eds.) (1916) Œuvres d’Henri Poincaré, Volume 2. Gauthier-Villars, Paris. Cited by: endnote 1.
• H. Poincaré (1882) Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 94, pp. 1038–1040. Cited by: endnote 1.
• H. P. d. Saint-Gervais (2016) Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich.