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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
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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

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References

REFERENCES

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