DOI QR코드

DOI QR Code

On the History of the Birth of Finsler Geometry at Göttingen

괴팅겐에서 핀슬러 기하가 탄생한 역사

  • Received : 2015.04.30
  • Accepted : 2015.06.22
  • Published : 2015.06.30

Abstract

Arrivals of Hilbert and Minkowski at $G\ddot{o}ttingen$ put mathematical science there in full flourish. They further extended its strong mathematical tradition of Gauss and Riemann. Though Riemann envisioned Finsler metric and gave an example of it in his inaugural lecture of 1854, Finsler geometry was officially named after Minkowski's academic grandson Finsler. His tool to generalize Riemannian geometry was the calculus of variations of which his advisor $Carath\acute{e}odory$ was a master. Another $G\ddot{o}ttingen$ graduate Busemann regraded Finsler geometry as a special case of geometry of metric spaces. He was a student of Courant who was a student of Hilbert. These figures all at $G\ddot{o}ttingen$ created and developed Finsler geometry in its early stages. In this paper, we investigate history of works on Finsler geometry contributed by these frontiers.

Keywords

References

  1. L. BLUMENTHAL, Distance Geometries, University of Missouri, 1938.
  2. H. BOERNER, CarathEodorys Eingang zur Variationsrechnung, Jahresbericht der Deutschen Mathematiker Vereinigung 56 (1953), 31-58.
  3. H. BUSEMANN, Metric Methods in Finsler Spaces and in the Foundations of Geometry, Princeton University Press, 1942.
  4. H. BUSEMANN, The Geometry of Finsler space, Bull. of the Amer. Math. Soc. 56(1) (1950), 5-16. https://doi.org/10.1090/S0002-9904-1950-09332-X
  5. H. BUSEMANN, The Geometry of Geodesics, Academic Press, 1955.
  6. C. Carath Eodory, Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Teubner, 1935.
  7. C. CARATHEODORY, Calculus of Variations and Partial Differential Equations of the first order, Part 1, Part 2, Holden-Day, 1965-1967.
  8. E. CARTAN, Sur les espaces de Finsler, C. R. Acad. Sci. Paris SEr. I Math. 196 (1933), 582-586.
  9. S. S. CHERN, Local Equivalence and Euclidean Connections in Finsler Spaces, Sci. Rep. Nat. Tsing Hua Univ. Ser. A 5 (1948), 95-121.
  10. S. S. CHERN, On Finsler Geometry, C. R. Acad. Sci. Paris SEr. I Math. 314 (1992), 757-761.
  11. S. S. CHERN, Finsler Geometry is just Riemannian Geometry without quadratic restriction, Notices of AMS 1 (1996), 959-963.
  12. S. S. CHERN, Riemannian Geometry as a special case of Finsler Geometry, Cont. Math. of the Amer. Math. Soc. 196 (1996), 51-58.
  13. A. EINSTEIN, Die Grundlage der allgemeinen Relativitatstheorie, Annalen der Physik 49 (1916), 769-822.
  14. P. FINSLER, Uber Kurven und Flachen in allgemeinen Raumen, Dissertation at the University of Gottingen, 1919.
  15. C. F. GAUSS, Disquisitiones generales circa superficies curvas, Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores 6 (1827), 99-146.
  16. S. KOBAYASHI, From Euclidean Geometry to Non-Euclidean Geometry, translated by D. Y. Won, Cheong Moon Gak, 1999. S. Kobayashi, 유클리드 기하에서 현대 기하로, 원대연 번역, 청문각, 1999.
  17. Mathematics Genealogy Project http://genealogy.math.ndsu.nodak.edu/index.php.
  18. H. MINKOWSKI, Geometrie der Zahlen (Geometry of Numbers), Teubner, 1910.
  19. B. RIEMANN, Uber die Hypothesen, welche der Geometrie zu Grunde liegen, Abhandlungen der Koniglichen Gesellschaft der Wissenschaften zu Gottingen 13 (1867), 1-15.
  20. H. RUND, Differential Geometry of Finsler Spaces, Springer, 1959.
  21. M. SPIVAK, A Comprehensive Introduction to Differential Geometry vol. II, 2nd Ed., Publish or Perish, 1999.
  22. J. TAYLOR, Parallelism and transversality in a sub-space of a general (Finsler) space, Ann. of Math. 28(2) (1927), 620-628.
  23. The MacTutor History of Mathematics Archive http://www-history.mcs.st-and.ac.uk/Biographies.
  24. R. THIELE, Hilbert's Twenty-Fourth Problem, American Mathematical Monthly 1 (2003), 1-24.
  25. Wikipedia http://en.wikipedia.org/wiki.
  26. WON D. Y., On the Development of Differential Geometry from mid 19C to early 20C by Christoffel, Ricci and Levi-Civita, Journal for History of Mathematics 28(2) (2015), 1-13. 원대연, 크리스토펠, 리치, 레비-치비타에 의한 19세기 중반부터 20세기 초반까지 미분기하학의 발전, Journal for History of Mathematics 28(2) (2015), 1-13. https://doi.org/10.14477/jhm.2015.28.1.001

Cited by

  1. 베어왈트에 의한 헝가리 데브레첸 핀슬러 기하학파의 형성의 역사 vol.31, pp.1, 2015, https://doi.org/10.14477/jhm.2018.31.1.037