## 4-52-11. H. Poincaré to Sophus Lie

[Ca. 09.10.1897]11endnote: 1 The manuscript bears an annotation in an unknown hand “[9.10.1897]”.

Mon cher Collègue,

Il y a quelques années lors de mon voyage à Leipzig, vous m’avez dit que vous aviez un moyen de construire toutes les surfaces qui sont doublement de translation.22endnote: 2 In early June, 1895, Poincaré visited Felix Klein in Göttingen, before travelling to Leipzig and Berlin. It is likely that while in Leipzig, Poincaré discussed surfaces and manifolds of double translation with Lie, as Poincaré had communicated a resumé (Poincaré, 1895b) of his memoir on this topic to the Paris Academy of Sciences on 4 February (Poincaré, 1895a). In the resumé, Poincaré found the occasion to recall Lie’s definition of a translation surface: “M. Lie a appelé surface de translation une surface dont les équations peuvent être mises sous la forme $x_{i}=f_{i}(t)+\phi_{i}(t^{\prime})\qquad(i=1,2,3),$ $x_{1}$, $x_{2}$, $x_{3}$ représentent les coordonnées rectangulaires d’un point dans l’espace et où $t$ et $t^{\prime}$ sont deux variables auxiliaires.” (Poincaré, 1895b, 241) The parametric curves of these equations are called directrix curves. Surfaces of double translation are surfaces of translation in two different ways, such that there are four families of directrix curves. Poincaré’s memoir, however, proposed an original method, and is concerned with several other topics, including the study of zeroes of Riemann’s theta function, as Adolf Hurwitz observed in his Jahrbuch review (Hurwitz, 1898).

Où avez-vous publié ce résultat.33endnote: 3 Lie published a long memoir on double translation surfaces, in which he commented on Poincaré’s resumé (1895b): “Recently in a note published in the first half of 1895 in the Compt. rend. Mr. Poincaré too has taken up the theory I established. Unfortunately, the distinguished writer whose achievements in other areas no one appreciates more than I do, neglected to acquaint himself sufficiently with my investigations. In any case, I cannot explain in any other way how he can be satisfied in his investigations into translation surfaces and translation manifolds with results that are subordinate, as very special cases, to my general theorems.” (Translated by S.A. Walter with original emphasis from Lie, 1896, 143) Lie followed up this dressing-down of Poincaré with a list of ten of his own related papers, published from 1872 to 1892. Was Poincaré unaware of Lie’s memoir of 1896? We recall that for the case of double translation manifolds, Lie did not provide a solution, such that, in light of Lié’s verbal boast in June, 1895, Poincaré’s question may have been read by Lie as a pointed barb. No further correspondence between the two men is extant; Lie died fourteen months after receiving this letter from Poincaré. In 1901, Poincaré developed a third approach to the problem (Poincaré, 1901), while additional contributions arrived from Scheffers (1904) and Darboux (1912). A generation passed before the Lie-Wirtinger theorem was established (Wirtinger, 1938), using what were later called Chow coordinates (Chow and Waerden, 1937). For references related to Poincaré’s paper of 1901 (up to 1925), see Garnier and Leray (1953, 37), and for an overview of early research on double translation surfaces and manifolds, see Little (1983) and Chern (1994).

Votre bien dévoué Collègue et ami,

Poincaré

ALS 2p. NB Ms. fol. 4507:A. Poincaré, Henri. Letter to Sophus Lie. National Library of Norway.

Time-stamp: " 3.11.2022 19:20"

### Notes

• 1 The manuscript bears an annotation in an unknown hand “[9.10.1897]”.
• 2 In early June, 1895, Poincaré visited Felix Klein in Göttingen, before travelling to Leipzig and Berlin. It is likely that while in Leipzig, Poincaré discussed surfaces and manifolds of double translation with Lie, as Poincaré had communicated a resumé (Poincaré, 1895b) of his memoir on this topic to the Paris Academy of Sciences on 4 February (Poincaré, 1895a). In the resumé, Poincaré found the occasion to recall Lie’s definition of a translation surface: “M. Lie a appelé surface de translation une surface dont les équations peuvent être mises sous la forme $x_{i}=f_{i}(t)+\phi_{i}(t^{\prime})\qquad(i=1,2,3),$ $x_{1}$, $x_{2}$, $x_{3}$ représentent les coordonnées rectangulaires d’un point dans l’espace et où $t$ et $t^{\prime}$ sont deux variables auxiliaires.” (Poincaré, 1895b, 241) The parametric curves of these equations are called directrix curves. Surfaces of double translation are surfaces of translation in two different ways, such that there are four families of directrix curves. Poincaré’s memoir, however, proposed an original method, and is concerned with several other topics, including the study of zeroes of Riemann’s theta function, as Adolf Hurwitz observed in his Jahrbuch review (Hurwitz, 1898).
• 3 Lie published a long memoir on double translation surfaces, in which he commented on Poincaré’s resumé (1895b): “Recently in a note published in the first half of 1895 in the Compt. rend. Mr. Poincaré too has taken up the theory I established. Unfortunately, the distinguished writer whose achievements in other areas no one appreciates more than I do, neglected to acquaint himself sufficiently with my investigations. In any case, I cannot explain in any other way how he can be satisfied in his investigations into translation surfaces and translation manifolds with results that are subordinate, as very special cases, to my general theorems.” (Translated by S.A. Walter with original emphasis from Lie, 1896, 143) Lie followed up this dressing-down of Poincaré with a list of ten of his own related papers, published from 1872 to 1892. Was Poincaré unaware of Lie’s memoir of 1896? We recall that for the case of double translation manifolds, Lie did not provide a solution, such that, in light of Lié’s verbal boast in June, 1895, Poincaré’s question may have been read by Lie as a pointed barb. No further correspondence between the two men is extant; Lie died fourteen months after receiving this letter from Poincaré. In 1901, Poincaré developed a third approach to the problem (Poincaré, 1901), while additional contributions arrived from Scheffers (1904) and Darboux (1912). A generation passed before the Lie-Wirtinger theorem was established (Wirtinger, 1938), using what were later called Chow coordinates (Chow and Waerden, 1937). For references related to Poincaré’s paper of 1901 (up to 1925), see Garnier and Leray (1953, 37), and for an overview of early research on double translation surfaces and manifolds, see Little (1983) and Chern (1994).

## Références

• S. Chern (1994) Sophus Lie and differential geometry. pp. 129–152. Cited by: endnote 3.
• W. Chow and B. L. v. d. Waerden (1937) Zur algebraischen Geometrie IX: Über zugeordnete Formen und algebraische Systeme von algebraischen Mannigfaltigkeiten. Mathematische Annalen 113, pp. 692–704. Cited by: endnote 3.
• G. Darboux (1912) Sur les surfaces de translation. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 155 (26), pp. 1449–1457. Cited by: endnote 3.
• R. Garnier and J. Leray (Eds.) (1953) Œuvres d’Henri Poincaré, Volume 6. Gauthier-Villars, Paris. Cited by: endnote 3.
• A. Hurwitz (1898) H. Poincaré, Remarques diverses sur les fonctions abéliennes. Jahrbuch über die Fortschritte der Mathematik 26, pp. 510–512. Cited by: endnote 2.
• O. A. Laudal and B. Jahren (Eds.) (1994) Proceedings of the Sophus Lie Memorial Conference, Oslo 1992. Scandanavian University Press, Oslo. Cited by: S. Chern (1994).
• S. Lie (1896) Die Theorie der Translationsflächen und das Abel’sche Theorem. Berichte über die Verhandlungen der königlich sächsischen Gesellschaft der Wissenschaften zu Leipzig, mathematisch-physische Klasse 48, pp. 141–198. Cited by: endnote 3.
• J. B. Little (1983) Translation manifolds and the converse of Abel’s theorem. Compositio Mathematica 40 (2), pp. 147–171. Cited by: endnote 3.
• H. Poincaré (1895a) Remarques diverses sur les fonctions abéliennes. Journal de mathématiques pures et appliquées 1 (3), pp. 219–314. Cited by: endnote 2.
• H. Poincaré (1895b) Sur les fonctions abéliennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 120 (5), pp. 239–243. Cited by: endnote 2, endnote 3.
• H. Poincaré (1901) Sur les surfaces de translation et les fonctions abéliennes. Bulletin de la Société mathématique de France 29, pp. 61–86. Cited by: endnote 3.
• G. Scheffers (1904) Das Abel’sche Theorem und das Lie’sche Theorem über Translationsflächen. Acta Mathematica 28, pp. 65–92. Cited by: endnote 3.
• W. Wirtinger (1938) Lie’s Translationsmannigfaltigkeiten und Abel’sche Integrale. Monatshefte für Mathematik und Physik 46 (1), pp. 384–431. Cited by: endnote 3.