Abstract
After DDK returned to India, he kept up his contacts with well-known European mathematicians such as T. Levi-Civita and É. Cartan, who also communicated his papers to professional journals. This paper appears to have been sent to Élie Cartan prior to publication, and the ensuing correspondence resulted in this paper by Kosambi and a note by Cartan being published back-to-back in Zeitschrift (see the next chapter). Along with a later paper by S.S. Chern in the Bulletin des Sciences Mathématiques, 63, 206–212 (1939) these papers lay the foundations of the Kosambi-Cartan-Chern theory.
Published in Mathematische Zeitschrift 37, 613–18 (1933).
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Notes
- 1.
D.D. Kosambi, The existence of a metric and the inverse variational problem, Bull. U.P. Acad. of Science, vol. 2. The main ideas of the investigation were set forth in a lecture to the Aligarh Mathematical Seminar on March 5, 1931. Some of the results of this paper have also been given in a note in the Rendiconti R. Accad. Dei Lincei 16 (1932), S. 410–415.
- 2.
In a personal letter, an extract of which is published after this paper. Mathematische Zeitschrift. 37.
- 3.
D.R. Davis in the Bull. Amer. Math. Soc. (1929), pp. 371–380. The equations given there would seen to be necessary but not sufficient.
- 4.
For a detailed bibliography of the subject, and in particular for references to the numerous papers of Berwald, I refer the reader to the article of Koschmieder, Jahresber. d. D.M.V. 40, pp.109–132. Other papers related to the present investigation are D.R. Davis, l. c. and Trans. Amer. Math. Soc. 33, p. 246 and J. Douglas, Ann. of Math. (II) 29.
- 5.
Note that for other parallelisms, where \(\gamma ^i_k\ne {1\over 2}\alpha ^i_{;k}\), we must have transformations that make both \(\gamma ^i_k\) and \(\alpha ^i_{;k}\) vanish simultaneously on the base, for reduction to the normal form; this is the general necessary and sufficient condition.
- 6.
See Eisenhart, Non-Riemannian Geometry, Am. Math. Soc. Coll. (1927), pp. 64–67.
- 7.
And the group, if solutions of the equation of variation define one as would be expected, will be valid in that neighborhood of a point within which no conjugate focus exists on any extremal through the point.
- 8.
Or else extra compatibility conditions are introduced \((\alpha ^i_{;k}-\gamma ^i_k)f_{;i}=0\).
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Kosambi, D.D. (2016). Parallelism and Path-Spaces. In: Ramaswamy, R. (eds) D.D. Kosambi. Springer, New Delhi. https://doi.org/10.1007/978-81-322-3676-4_7
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