## 7-2-55. Felix Klein to H. Poincaré, English translation

Leipzig, 4 December 1881

Sophienstraße 10/II

Dear Sir!

For my part, I have meanwhile written up a little treatise on ‘‘Riemann’s theory’’ which may be of interest to you since it presents a version of the concept of a Riemann surface that I believe R. himself actually worked on.22endnote: 2 Klein (1882), reed. Klein (1923, 499–573). Perhaps Mr. Brunel has told you of this. I have also been busy lately with different existence proofs designed to replace Dirichlet’s principle, and I am convinced that the methods expounded by Schwarz in the Berliner Monatsberichten, 1870, p. 767 et seqq. in any case suffice completely to yield, for example, the general theorem I wrote to you about once or twice this past summer.33endnote: 3 Schwarz (1870).

With the greatest respect,

F. Klein

PTrL. Translated by R. Burns from the original German (§ 4-47-10), in Saint-Gervais (2016), with slight revisions by S.A. Walter. See also the French translation (§ 7-2-36).

Time-stamp: "28.04.2021 17:33"

### Notes

• 1 Poincaré (1881), reed. Nörlund and Lebon (1916, 29–31). “[T]hat every linear differential equation with algebraic coefficients is integrable by means of zetafuchsian functions and that the coordinates of any algebraic curve whatever can be expressed via fuchsian functions of an auxiliary variable.”
• 2 Klein (1882), reed. Klein (1923, 499–573).
• 3 Schwarz (1870).

## References

• F. Klein (1882) Ueber Riemann’s Theorie der algebraischen Functionen und ihrer Integrale. Teubner, Leipzig. Cited by: endnote 2.
• F. Klein (1923) Gesammelte mathematische Abhandlungen, Volume 3. Springer, Berlin. Cited by: endnote 2.
• N. E. Nörlund and E. Lebon (Eds.) (1916) Œuvres d’Henri Poincaré, Volume 2. Gauthier-Villars, Paris. Cited by: endnote 1.
• H. Poincaré (1881) Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 93, pp. 301–303. Cited by: endnote 1.
• H. P. d. Saint-Gervais (2016) Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich.
• H. A. Schwarz (1870) Ueber die Integration der partiellen Differentialgleichung $\frac{\partial^{2}u}{\partial x^{2}}{\partial^{2}u}{\partial y^{2}}=0$ unter vorgeschriebenen Grenz- und Unstetigkeitsbedingungen. Monatsberichte der königliche Akademie der Wissenschaften zu Berlin, pp. 767–795. Cited by: endnote 3.