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On the existence of metric differential geometries based on the notion of area

Published online by Cambridge University Press:  24 October 2008

F. Brickell
F. Brickell
Affiliation:
University CollegeSouthampton
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Extract

The problem of constructing an n-dimensional metric differential geometry based on the idea of a two-dimensional area has given rise to several publications, notably by A. Kawaguchi and S. Hokari (1), E. T. Davies (2), and R. Debever (3). In this geometry the area of a two-dimensional plane element is defined by a fundamental function L(xi, uhk), where the xi are point coordinates and the uhk are the coordinates of the simple bivector representing the plane element. L is supposed to be a positive homogeneous function of the first degree with respect to the variables uij, and to possess continuous partial derivatives up to and including those of the fourth order. With these assumptions the problem of the construction of the metric differential geometry splits into two problems; the first of these is the problem of constructing a metric tensor gij(xr, uhk), and the second is the problem of constructing an affine connexion. We deal with the first problem only in this paper.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 1 , January 1950 , pp. 67 - 72
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

REFERENCES

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