Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-20T04:32:28.735Z Has data issue: false hasContentIssue false
  • English
  • Français

A Generalization of Finsler Geometry

Published online by Cambridge University Press:  20 November 2018

J. R. Vanstone
J. R. Vanstone*
Affiliation:
University of Natal and University of Toronto
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Modern differential geometry may be said to date from Riemann's famous lecture of 1854 (9), in which a distance function of the form F(xi, dxi) = (γij(x)dxidxj½ was proposed. The applications of the consequent geometry were many and varied. Examples are Synge's geometrization of mechanics (15), Riesz’ approach to linear elliptic partial differential equations (10), and the well-known general theory of relativity of Einstein.

Meanwhile the results of Caratheodory (4) in the calculus of variations led Finsler in 1918 to introduce a generalization of the Riemannian metric function (6). The geometry which arose was more fully developed by Berwald (2) and Synge (14) about 1925 and later by Cartan (5), Busemann, and Rund. It was then possible to extend the applications of Riemannian geometry.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 87 - 112
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Albert, A., Modern higher algebra (Chicago, 1937).Google Scholar
2. Berwald, L., Untersuchung der Kriimmung allgemeiner metrischer Rdume auf Grund des in ihnen herrschenden P arallelismus, Math. Z., 25 (1926), 40–73.Google Scholar
3. Berwald, L., Ueber Finslersche und Cartansche Géométrie III. Two-dimensional Finsler spaces with rectilinear extremals, Ann. Math. (2), 42 (1941), 84–112.Google Scholar
4. Caratheodory, C., Ueber die starken Maxima und Minima bei einfachen Integralen, Math. Ann., 62 (1906), 449–503.Google Scholar
5. Cartan, E., Les espaces de Finsler, Actualités, 79 (Paris 1934).Google Scholar
6. Finsler, P., Ueber Kurven und Fldchen in allgemeinen Ràumen (Dissertation, Gôttingen, 1918). Unverânderter Nachdruck (Basel: Birkhâuser, 1951).Google Scholar
7. Moór, A., Entwicklung einer Géométrie der allgemeinen metrischen Linienelementraume, Acta Sci. Math. Szeged, 17 (1956), 85–120.Google Scholar
8. Moór, A., Ueber die Autoparallele Abweichung in Allgemeinen metrischen Linienelemenfràumen, Publicationes Mathematicae, Debrecen, 5 (1957), 103–118.Google Scholar
9. Riemann, B., Ueber die Hypothesen, welche der Geometric zugrunde liegen, Habilitationsvortrag 1854, Ges. math. Werke., 272–287.Google Scholar
10. Riesz, M., L'intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math., 81 (1949), 1–223.Google Scholar
11. Rund, H., Die Hamiltonsche Funktion bei allgemeinen dynamischen Systemen, Arch. Math., 3 (1952), 207–215.Google Scholar
12. Rund, H., Differential Geometry of Finsler Spaces (Berlin: Springer-Verlag, 1959).Google Scholar
13. Rund, H., The scalar form of Jacobïs equations in the calculus of variations, Ann. Math, pura appl. (4), 35 (1953), 183–202.Google Scholar
14. Synge, J., A generalization of the Riemannian line-element, Trans. Amer. Math. Soc, 27 (1925), 61–67.Google Scholar
15. Synge, J., The Geometry of Dynamics, Phil. Trans. Roy. Soc. London (A), 226 (1926), 31–106.Google Scholar