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A Study of Tensors which Characterize a Hypersurface of a Finsler Space

Published online by Cambridge University Press:  20 November 2018

Gillian M. Brown
Gillian M. Brown*
Affiliation:
University of Toronto, Toronto, Ontario; University of Canterbury, New Zealand
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The literature on Finsler geometry contains more than one definition for the normal curvature vector of a hypersurface and for coefficients of the second fundamental form; see Berwald (1), Davies (3), and Rund (5). In the first case this situation has arisen from the basically different approach to the subject adopted by the authors; Davies, following the locally Euclidean school and Rund the locally Minkowskian theory. In both cases, a comparison of the definitions shows that they are linked by expressions in a vector Mαwhich was introduced in the paper by Rund (7).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 1025 - 1036
Copyright
Copyright © Canadian Mathematical Society 1968

Footnotes

This paper comprises a portion of a doctoral thesis submitted to the University of Toronto.

References

1. Berwald, L., Uber die Hauptkrûmmungen einer Flàche im dreidimensionalen Finslerschen Raum, Mh. Math. Phys. 43 (1936), 1–14.Google Scholar
2. Berwald, L., Untersuchung der Kriimmung allgemeiner metrischer Ràume auf Grund des in ihnen herrschenden Parallelismus, Math. Z. 25 (1926), 40–73.Google Scholar
3. Davies, E. T., Subspaces of a Finsler space, Proc. London Math. Soc. (2) 49 (1947), 19–39.Google Scholar
4. Deicke, A., Ùber die Finsler-Ràume mit Ai - 0, Arch. Math. 4 (1953), 45–51.Google Scholar
5. Rund, H., Differential geometry of Finsler spaces (Springer-Verlag, Berlin, 1959).10.1007/978-3-642-51610-8CrossRefGoogle Scholar
6. Rund, H., Curvature properties of hyper surf aces of Finsler and Minkowskian spaces, Tensor N.S.) 14 (1963), 226–244.Google Scholar
7. Rund, H., Intrinsic and induced curvature theories of subspaces of a Finsler space, Tensor (N.S.) 16 (1965), 294–312.Google Scholar