×

Variations on a theme: Clifford’s parallelism in elliptic space. (English) Zbl 1331.51017

Author’s abstract: In 1873, W. K. Clifford introduced a notion of parallelism in the three- dimensional elliptic space that, quite surprisingly, exhibits almost all properties of Euclidean parallelism in ordinary space. The purpose of this paper is to describe the genesis of this notion in Clifford’s works and to provide a historical analysis of its reception in the investigations of F. Klein, L. Bianchi, G. Fubini, and E. Bortolotti. Special emphasis is placed upon the important role that Clifford’s parallelism played in the development of the theory of connections.

MSC:

51M10 Hyperbolic and elliptic geometries (general) and generalizations
53B21 Methods of local Riemannian geometry
51-03 History of geometry
01A55 History of mathematics in the 19th century
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ball, R.S. 1876. The theory of screws: A study in the dynamics of a rigid body. Dublin: Hodges, Foster and Co. · JFM 08.0599.01
[2] Ball, R.S. 1881. Certain problems in the dynamics of a rigid system moving in elliptic space. Transactions of the Royal Irish Academy 28: 159-184.
[3] Beltrami, E. 1865. Risoluzione del problema: “riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette”. Annali di Matematica pura ed applicata 7: 185-204. Also in (Beltrami Opere Matematiche, I, 262-280). · doi:10.1007/BF03198517
[4] Beltrami, E. 1868. Saggio di interpretazione della geometria non-euclidea. Giornale di Matematiche 6: 284-312. Also in (Beltrami Opere Matematiche, I, 374-405). · JFM 01.0275.02
[5] Beltrami, E. 1868-1869. Teoria fondamentale degli spazi a curvatura costante, Annali di Matematica pura ed applicata, 2: 232-255. Also in (Beltrami Opere Matematiche, I, 406-429). · JFM 23.0527.01
[6] Beltrami, E. 1902-1920. Opere Matematiche, 4 Volumes, Hoepli, Milano.
[7] Betten, D., and R. Riesinger. 2012. Clifford parallelism: Old and new definitions, and their use. Journal of Geometry 103: 31-73. · Zbl 1258.51001 · doi:10.1007/s00022-012-0118-2
[8] Bianchi, L. 1895. Sulle superficie a curvatura nulla negli spazi di curvatura costante. Atti dell’Accademia delle Scienze di Torino 30: 475-487. Also in (Bianchi Opere, VIII, 256-265). · JFM 26.0700.01
[9] Bianchi, L. 1896. Sulle superficie a curvatura nulla in geoemtria ellittica. Annali di Matematica Pura ed Applicata 25(2): 93-129. Also in (Bianchi Opere, VIII, 266-301). · JFM 27.0370.02 · doi:10.1007/BF02419524
[10] Bianchi, L. 1902. Lezioni di Geometria Differenziale, vol. 1. Pisa: Spoerri. · JFM 25.1165.01
[11] Bianchi, L. 1904. Lezioni di Geometria Analitica. Pisa: Spoerri. · JFM 47.0583.11
[12] Bianchi, L. 1922. Sul parallelismo vincolato di Levi-Civita nella metrica degli spazi curvi. Rendiconti della Accademia delle Scienze di Napoli 28: 150-171. Also in (Bianchi Opere, X, 43-64). · JFM 48.0850.04
[13] Bianchi, L. 1952-1959. Opere, vol: 11. Roma: Edizioni Cremonese. · JFM 52.0422.04
[14] Bompiani, E. 1942. Enea Bortolotti. Rendiconti di Matematica e delle sue Applicazioni 3: 241-281. · JFM 68.0018.02
[15] Bonola, R. 1912. Non-euclidean geometry. Chicago: The Open Court Publishing Company. · JFM 43.0557.02
[16] Bortolotti, E. 1924-1925. (Enea), Parallelismo assoluto e vincolato negli \[S_3\] S3 a curvatura costante ed estensione alle \[V_3\] V3 qualunque, Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti, 84: 821-858. · JFM 16.0465.01
[17] Bortolotti, E. 1927. Parallelismi assoluti nelle \[V_n\] Vn riemanniane. Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti 84: 455-465. · JFM 53.0688.02
[18] Bortolotti, E. 1930. On parallelisms and teleparallelisms in curved space. Journal of London Mathematical Society 5: 242-248. · JFM 56.0616.02 · doi:10.1112/jlms/s1-5.4.242
[19] Bortolotti, E. 1935. Lezioni di Geoemtria Superiore. Firenze: Poligrafica Universitaria.
[20] Buchheim, A. 1883. On the theory of screws in elliptic space. Proceedings of the London Mathematical Society 15: 83-98. (Read January 1884). · JFM 16.0465.01 · doi:10.1112/plms/s1-15.1.83
[21] Buchheim, A. 1884. On the theory of screws in elliptic space (supplementary note). Proceedings of the London Mathematical Society 16: 15-27. · doi:10.1112/plms/s1-16.1.15
[22] Buchheim, A. 1885. A memoir on biquaternions. American Journal of Mathematics 7: 293-326. · JFM 17.0678.01 · doi:10.2307/2369176
[23] Cartan, E., and J.A. Schouten. 1926. On the geometry of the group-manifold of simple and semi-simple groups. Proceedings of the Royal Academy Amsterdam 29: 803-815. Also in (Cartan OEuvres, Part 1, vol. II; 573-585). · JFM 52.0422.04
[24] Cartan, É. 1924. Les récentes généralisations de la notion d’espace. Bulletin des Sciences Mathématiques 48: 294-320. Also in (Cartan OEuvres, Part 3, vol. I; 863-889). · JFM 50.0589.01
[25] Cartan, É. 1952-1955. Œuvres Complètes. Paris: Gauthier-Villars. · Zbl 0049.30302
[26] Castelnuovo, G. 1904. Lezioni di Geometria Analitica e Proiettiva, Roma-Milano. · JFM 35.0559.04
[27] Cayley, A. 1859. A sixth memoir upon quantics. Philosophical Transactions of the Royal Society of London 149: 61-90. Also in (Cayley Mathematical Papers, II, 561-592). · JFM 01.0275.02
[28] Cayley, A. 1889-1897. Collected mathematical papers, 13 Volumes, Cambridge.
[29] Clifford, W.K. 1873. Preliminary sketch of biquaternions. Proceedings of the London Mathematical Society, 4, 64-65, 381-395. Also in (Clifford Mathematical Papers, 181-200).
[30] Clifford, W.K. 1882. Mathematical papers. London: MacMillan.
[31] Conforto, F. 1948. Enea Bortolotti. Enciclopedia Italiana Treccani, II Appendice. Rome: Treccani.
[32] Coxeter, H.S.M. 1998. Non-euclidean geometry, 6th ed. Washington D.C.: The Mathemamatical Association of America. · Zbl 0909.51003
[33] Fubini, G. 1900. Il parallelismo di Clifford negli spazi ellittici. Tipografia Successori, Pisa: Tesi di Laurea. · JFM 35.0668.03
[34] Giering, O. 1982. Vorlesungen über höhere Geometrie. Braunschweig-Wiesbaden: Vieweg. · Zbl 0493.51001 · doi:10.1007/978-3-322-83552-9
[35] Klein, F. 1871. Notiz, betreffend den Zusammenhang der Liniengeometrie mit der Mechanik starrer Körper. Math. Ann. 4: 403-415. Also in (Klein Ges. Math. Abh., I, 226-240). · JFM 03.0446.02 · doi:10.1007/BF01455075
[36] Klein, F. 1871. Über die sogennante Nicht-Euklidische Geometrie. Math. Ann. 4: 573-625. Also in (Klein Ges. Math. Abh., I, 254-305). · JFM 03.0231.02 · doi:10.1007/BF02100583
[37] Klein, F. 1890. Zur Nicht-Euklidische Geometrie. Math. Ann. 37: 544-572. Also in (Klein Ges. Math. Abh., I, 353-383). · JFM 22.0535.01 · doi:10.1007/BF01724772
[38] Klein, F. 1921-1923. Gesammelte Mathematische Abhandlungen, 3 volumes. Berlin: Springer.
[39] Klein, F. 1928. Vorlesungen über nicht-Euklidische Geometrie, Berlin. · JFM 54.0593.01
[40] Levi-Civita, T. 1917. Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura Riemanniana. Rendiconti del Circolo Matematico di Palermo 42: 173-205. Also in (Levi-Civita, IV, 1-39). · JFM 46.1125.02 · doi:10.1007/BF03014898
[41] Levi-Civita, T. 1954-1973. Opere Matematiche, 6 Volumes, Zanichelli, Bologna. · Zbl 0055.40104
[42] Plücker, J. 1866. Fundamental views regarding mechanics. Philosophical Transactions of the London Royal Society 156: 361-380. · doi:10.1098/rstl.1866.0016
[43] Reich, K. 1992. Levi-Civitasche Parallelverschiebung, affiner Zusammenhang, Über tragungsprinzip, 1916-1917-1922/1923. Archive for History of Exact Sciences 44: 77-105. · Zbl 0767.01023 · doi:10.1007/BF00379682
[44] Rowe, D.E. 1989. The early geometrical works of Sophus Lie and Felix Klein. In The history of modern mathematics I, eds. D.E. Rowe and J. McCleary, 210-273. New York. · JFM 52.0422.04
[45] Seidenberg, A. 1962. Lectures in projective geometry. New York: Dover. · Zbl 0121.37705
[46] Study, E. 1891. Von den Bewegungen und Umlegungen, I und II Abhandlung. Math. Ann. 39: 441-566. · JFM 23.0527.01 · doi:10.1007/BF01199824
[47] Study, E. 1913. Grundlagen und Ziele der analytischen Kinematik. Sitzber. d. Berl. math. Ges. 13: 36-60. · JFM 44.0791.03
[48] Vaney, F. 1929. Le parallélisme absolu dans les espaces elliptiques réels à 3 et à 7 dimensions et le principe de trialité dans l’espace elliptique à 7 dimensions. Paris: Gauthier-Villars.
[49] Zacharias, M. 1913. Elementargeometrie und elementare nichteuklidische Geometrie in synthetischer Behandlung, Enzyklopädie der Mathematischen Wissenschaften mit Einschluß ihrer Anwendungen, Bd. 3-1-2, 862-1162. Leipzig: B.G. Teubner Verlag.
[50] Ziegler, R. 1985. Die Geschichte der geometrischen Mechanik im 19. Jahrhundert. Stuttgart: Steiner. · Zbl 0579.01012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.