Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T08:13:43.235Z Has data issue: false hasContentIssue false
  • English
  • Français

Angular Measure and Integral Curvature

Published online by Cambridge University Press:  20 November 2018

Herbert Busemann
Herbert Busemann*
Affiliation:
University of Southern California
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.

The Gauss-Bonnet Theorem leads through well known arguments to the fact that the integral curvature of a two-dimensional closed orientable manifold M of genus p equals 4π(1 — p). This implies, for instance, that the Gauss curvature K can neither be everywhere positive nor everywhere negative, if M is homeomorphic to a torus.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 3 , 01 June 1949 , pp. 279 - 296
Copyright
Copyright © Canadian Mathematical Society 1949

References

[1] Busemann, H., “Metric Methods in Finsler Spaces and in the Foundations of Geometry,” Ann. Math. Studies No. 8 (Princeton, 1942).Google Scholar
[2] Busemann, H., “Local Metric Geometry,” Trans. Am. Math. Soc.,vo. 56 56 (1944), 200-274.Google Scholar
[3] Busemann, H., “Spaces with Non-positive Curvature,” Acta Math. Google Scholar
[4] Cohn-Vossen, S., “Kürzeste Wege und Totalkriimmung auf Flächen,” Comp. Math., vol. 2 (1935), 69-133.Google Scholar
[5] Cohn-Vossen, S., “Totalkrümmung und geodätische Linien auf einfach zusammenhängenden, offenen, vollständigen Flächenstücken,” Mat. Sbornik, N. S., vol. 1 (1936), 139–164.Google Scholar
[6] Hadamard, J., “Les surfaces a courbures opposées et leur lignes géodésiques,” Jour. Math. Pur. Appl., 5th series, vol. 4 (1898), 27-73.Google Scholar
[7] Jessen, B., Abstrakt Maal-og Integralteori (Copenhagen, 1947).Google Scholar
[8] Kerékjártó, B.v., Vorlesungen über Topologie I (Berlin, 1923).Google Scholar
[9] Seifert, H. and Threlfall, W., Lehrbuch der Topologie (Leipzig, 1934).Google Scholar