Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-29T11:27:23.541Z Has data issue: false hasContentIssue false
  • English
  • Français

The theory of surfaces in a geometry based on the notion of area

Published online by Cambridge University Press:  24 October 2008

E. T. Davies
E. T. Davies
Affiliation:
King's College, Strand London, W.C.2
Get access Rights & Permissions [Opens in a new window]

Extract

In his tract on spaces based on the notion of area, Cartan (1) gives a theory of curves and of surfaces immersed in his 3-dimensional space. A theory of curves is possible by associating with each point of the curve the 2-dimensional orientation of the contravariant bivector (or the corresponding covariant vector density of weight − 1) of support which is normal to the curve at that point. This association is possible, since there is a unique normal 2-direction at every point of a curve in a 3-dimensional space. If the surrounding space has more than 3 dimensions, however, and the element of support is still a bivector, then a theory of curves becomes impossible since no unique normal or tangential 2-direction can be associated with the curve at every point. A theory of surfaces is still possible because a surface has a unique tangential 2-direction which can be taken to be the 2-direction of the bivector of support. It is the object of this note to consider what subspaces can be admitted by a space when the number of dimensions is n > 3.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 43 , Issue 3 , July 1947 , pp. 307 - 313
Copyright
Copyright © Cambridge Philosophical Society 1947

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Cartan, E., Lea espaces métriques fondés sur la notion d'aire, Actualiée Sci. 72 (Paris, 1933).Google Scholar
(2)Cartan, E., Lea espaces de Finsler, Actualités Sci. 79 (1934).Google Scholar
(3)Davies, E. T., J. London Math. Soc. 20 (1945), 163–171.Google Scholar
(4)Davies, E. T.Proc. London Math. Soc. Ser. 2, 49 (1945), 19–40.Google Scholar
(5)Davies, E. T.Quart. J. Math. (Oxford series), 16 (1945), 22–31.CrossRefGoogle Scholar
(6)Schouten, J. A. and Struik, D. J.Einfüihrung in die neueren Methoden der Differentialgeometrie, Gröningen, 1 (1935), 2 (1938).Google Scholar
(7)Schouten, J. A.Der Ricci-Kalkul (Berlin, 1924).Google Scholar
(8)Bianchi, L.Lezioni di geometria differenziale, Zanichelli, 1 (1924), part I.Google Scholar
(9)Bortolotti, E.Rend. R. 1st. Lombardo, 64 (1931), 1–23.Google Scholar
(10)Duschek, A. and Mayer, W.Lehrbuch der Differentialgeometrie, Teubner, 2 (1930).Google Scholar