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 the topic of double translation surfaces and manifolds 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 xi=fi(t)+ϕi(t)  (i=1,2,3),x_{i}=f_{i}(t)+\phi_{i}(t^{\prime})\qquad(i=1,2,3), x1x_{1}, x2x_{2}, x3x_{3} représentent les coordonnées rectangulaires d’un point dans l’espace et où tt et tt^{\prime} sont deux variables auxiliaires.” (Poincaré, 1895b, 241) 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 review.

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. Poıncaré 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 knowledge claim 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). 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 (1983).

Votre bien dévoué Collègue [illegible]

Poincaré

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

Time-stamp: "10.08.2022 00:48"

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 the topic of double translation surfaces and manifolds 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 xi=fi(t)+ϕi(t)  (i=1,2,3),x_{i}=f_{i}(t)+\phi_{i}(t^{\prime})\qquad(i=1,2,3), x1x_{1}, x2x_{2}, x3x_{3} représentent les coordonnées rectangulaires d’un point dans l’espace et où tt et tt^{\prime} sont deux variables auxiliaires.” (Poincaré, 1895b, 241) 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 review.
  • 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. Poıncaré 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 knowledge claim 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). 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 (1983).

Références

  • F. E. Browder (Ed.) (1983) The Mathematical Heritage of Henri Poincaré, Volume 1. American Mathematical Society, Providence. Cited by: S. Chern (1983).
  • S. Chern (1983) Web geometry. See The Mathematical Heritage of Henri Poincaré, Volume 1, Browder, Proceedings of Symposia in Pure Mathematics, Vol. 39, pp. 3–10. Link, Document 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. Link Cited by: endnote 3.
  • R. Garnier and J. Leray (Eds.) (1953) Œuvres d’Henri Poincaré, Volume 6. Gauthier-Villars, Paris. Link Cited by: endnote 3.
  • 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. Link Cited by: endnote 3.
  • J. B. Little (1983) Translation manifolds and the converse of Abel’s theorem. Compositio Mathematica 40 (2), pp. 147–171. Link Cited by: endnote 3.
  • H. Poincaré (1895a) Remarques diverses sur les fonctions abéliennes. Journal de mathématiques pures et appliquées 1 (3), pp. 219–314. Link 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. Link 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. Link Cited by: endnote 3.
  • G. Scheffers (1904) Das Abel’sche Theorem und das Lie’sche Theorem über Translationsflächen. Acta Mathematica 28, pp. 65–92. Link, Document Cited by: endnote 3.
  • W. Wirtinger (1938) Lie’s Translationsmannigfaltigkeiten und Abel’sche Integrale. Monatshefte für Mathematik und Physik 46 (1), pp. 384–431. Link, Document Cited by: endnote 3.