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Felix Klein, Sophus Lie, contact transformations, and connexes

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Abstract

Much of the mathematics with which Felix Klein and Sophus Lie are now associated (Klein’s Erlangen Program and Lie’s theory of transformation groups) is rooted in ideas they developed in their early work: the consideration of geometric objects or properties preserved by systems of transformations. As early as 1870, Lie studied particular examples of what he later called contact transformations, which preserve tangency and which came to play a crucial role in his systematic study of transformation groups and differential equations. This note examines Klein’s efforts in the 1870s to interpret contact transformations in terms of connexes and traces that interpretation (which included a false assumption) over the decades that follow. The analysis passes from Klein’s letters to Lie through Lindemann’s edition of Clebsch’s lectures on geometry in 1876, Lie’s criticism of it in his treatise on transformation groups in 1893, and the careful development of that interpretation by Dohmen, a student of Engel, in his 1905 dissertation. The now-obscure notion of connexes and its relation to Lie’s line elements and surface elements are discussed here in some detail.

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Notes

  1. For a description of the Berlin Mathematics Club, see Grötschel (2008).

  2. For a discussion of Klein’s and Lie’s work during this early period, and in particular of their influence on each other’s thinking, see Hawkins (1989), Rowe (1989), and Rowe (2023).

  3. Letter from Klein to Lie (Klein), 6 October 1871. All translations from the Klein/Lie letters cited here are by the author.

  4. Letter from Klein to Lie (Klein), 5 December 1871.

  5. Letter from Klein to Lie (Klein), 29 June 1872.

  6. A footnote to Lie (1872a), dated 11 October 1872, refers to an oral communication from Clebsch.

  7. Letter from Klein to Lie (Klein), 18 May 1872.

  8. Letter from Klein to Lie (Klein), 9 February 1872.

  9. Letter from Klein to Lie (Klein), 15 December 1872.

  10. Letter from Klein to Lie (Klein), 21 November 1872.

  11. Letter from Klein to Lie (Klein), 4 February 1873.

  12. Letter from Klein to Lie (Klein), 10 February 1873.

  13. Letter from Klein to Lie (Klein), 4 May 1873.

  14. Letter from Klein to Lie (Klein), 20 July 1875.

  15. Letter from Klein to Max Noether, 1 March 1899, cited in Rowe and Tobies (1990, p. 8). Klein destroyed all his correspondence, not just that with Lie.

  16. Letter from Klein to Lie (Klein), 27 January 1876.

  17. Several French geometers, including Darboux, had earlier developed a somewhat different geometry based on spheres, which may well have influenced Lie’s thinking. See Rowe (1989, Sects. 6 and 7).

  18. The first condition is explicit in his definition; the second is explicit in a number of his proofs. See, for example, Lie (1875a, p. 220) or Lie (1890, pp. 45–46, 115, 120.).

  19. Lie (1873b) and Lie (1890, p. 114).

  20. See Hawkins (1991) and Hawkins (2000) (particularly Chapters 1–3).

  21. The paper Clebsch (1872) appeared in Gött. Nach. and was reprinted essentially unchanged in Math. Ann., immediately after his obituary.

  22. Letter from Klein to Lie (Klein), 20 July 1875. In the Erlangen Program, he had used the term connex-element for a different configuration in space.

  23. Letter from Klein to Lie (Klein), 10 February 1873. See also Clebsch and Lindemann (1876, p. 1021).

  24. Lie used both “dimension” and “order” in the 1870s and “order” in Transformationsgruppen, twenty years later; Klein used “degree” in his letters and also in his 1892/3 lectures; Lindemann used both “dimension” and “degree” in the Clebsch lectures.

  25. Letter from Klein to Lie (Klein), 10 February 1873.

  26. This paragraph is illustrative but is not a proof that the components of any homogeneous contact transformation have the asserted homogeneity properties. This argument is used in Lie (1890, Section 37) to show that if a contact transformation has the form (11), then the \(X_i\)’s are homogeneous of degree 0 and the \(P_i\)’s of degree 1 in the \(p_i\)’s. But (11) is derived from (10) by showing that if a contact transformation of the form (10) is homogeneous, then the conditions \([Az + \Omega ,\, X_i]=0\) and \([P_i,\, Az+\Omega ] = 0\) force \(\Omega \) to be constant.

  27. Lie gave no details. A proof was given in 1905 by F. J. Dohmen, a student of Engel, in his dissertation (Dohmen 1905, pp. 2–32).

  28. Letter from Klein to Lie (Klein), 23 July 1874.

  29. Letter from Klein to Lie (Klein), 20 July 1875. Klein is clearly referring to Lie (1875b) because he discusses in detail a function H introduced there.

  30. These lectures were edited and published in 1926 (Klein 1926).

  31. (Klein 1893, p. 547), (Klein 1926, p. 298). A footnote on the same page in Klein (1926) also cites Lie (1893, p. 530 ff.), with no further comment.

  32. (Klein 1893, p. 537), (Klein 1926, p. 297).

  33. (Klein 1893, p. 466), (Klein 1926, p. 242).

  34. These troubles may have been early symptoms of the pernicious anemia that was not diagnosed until 1898 and of which he died in 1899. See Fritzsche (1991).

  35. These two letters are cited in Rowe (1988). Tobies’s biography (Tobies 2019) of Klein, Stubhaug’s biography (Stubhaug 2002) of Lie, and the articles Rowe (1988) and Rowe (2023) by Rowe discuss Lie’s difficult relationships with Klein and other mathematicians during his time in Leipzig.

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Acknowledgements

Permission to use the letters (Klein) was granted by Morten Eide, for the family of Torger Holtsmark. The author gratefully acknowledges that much of this work was done during the author’s affiliation with the Virginia Tech Mathematics Department, and in response to questions raised by David Rowe in reading the letters. The author is grateful to the referee for many helpful comments that led to substantial improvements in this paper.

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    L. D. Kay

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Kay, L.D. Felix Klein, Sophus Lie, contact transformations, and connexes. Arch. Hist. Exact Sci. 77, 373–391 (2023). https://doi.org/10.1007/s00407-023-00305-1

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