4-52-11. H. Poincaré à Sophus Lie

[Ca. 09.10.1897]

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.

Où avez-vous publié ce résultat.

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: "10.04.2024 13:20"

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 $$x_{i}=f_{i}(t)+\phi_{i}(t^{\prime})\qquad(i=1,2,3),$$ où $x_{1}$, $x_{2}$, $x_{3}$ 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; Dieudonné, 1985). 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).