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Subspaces of a Generalized Metric Space

Published online by Cambridge University Press:  20 November 2018

H. A. Eliopoulos
H. A. Eliopoulos*
Affiliation:
Assumption University of Windsor
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In a paper published in 1956, Rund (4) developed the differential geometry of a hypersurface of n— 1 dimensions imbedded in a Finsler space of n dimensions, considered as locally Minkowskian.

The purpose of the present paper is to provide an extension of the results of (4) and thus develop a theory for the case of m-dimensional subspaces imbedded in a generalized (Finsler) metric space.

We consider an n-dimensional differentiable manifold Xn and we restrict our attention to a suitably chosen co-ordinate neighbourhood of Xn in which a co-ordinate system xi (i= 1, 2, … , n), is defined. A system of equations of the type xi = xi(t) defines a curve C of Xn the tangent vector dxi/dt of which is denoted by xi.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 235 - 255
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Coxeter, H. S. M., The real projective plane (Cambridge, 1955).Google Scholar
2. Eisenhart, L. P., Riemannian Geometry (Princeton, 1949).Google Scholar
3. Eliopoulos, H. A., Methods of generalized metric geometry with applications to mathematical physics, Ph.D. thesis, University of Toronto, August, 1956.Google Scholar
4. Rund, H., Hypersurfaces of a Finsler space, Can. J. Math., 8 (1956), 487–503.Google Scholar
5. Rund, H. Ueber die Parallelverschiebung in Finslerschen Raumen, Math. Z., 54 (1951), 115–128.Google Scholar
6. Rund, H. Qn the analytical properties of curvature tensors in Finsler spaces, Math. Ann., 127 (1954), 82–104.Google Scholar
7. Tucker, A. W., On generalized covariant differentiation, Ann. Math., 82 (1931), 451–60.Google Scholar