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Autoparallel Deviation in the Geometry of Lyra

Published online by Cambridge University Press:  20 November 2018

J. R. Vanstone
J. R. Vanstone*
Affiliation:
University of Toronto
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One of the fruitful tools for examining the properties of a Riemannian manifold is the study of “geodesic deviation”. The manner in which a vector, representing the displacement between points on two neighbouring geodesies, behaves gives an indication of the difference between the manifold and an Euclidean space. The study is essentially a geometrical approach to the second variation of the lengthintegral in the calculus of variations [1]. Similar considerations apply in the geometry of Lyra [2] but as we shall see, appropriate analytical modifications must be made. The approach given here is modelled after that of Rund [3] which was originally designed to deal with a Finsler manifold but which applies equally well to the present case.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 3 , Issue 3 , 01 September 1960 , pp. 263 - 271
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Blaschke, W., Differential Geometry, (New York, 1945) p. 212, et seq.Google Scholar
2. Lyra, G., Über eine Modifikation der riemannschen Geometrie, Math, Z. 54 (1951), 52-64.Google Scholar
3. Rund, H., The Differential Geometry of Finsler Spaces, Springer-Verlag (Berlin-Gottingen-Heidelberg, 1959), 111-119.Google Scholar
4. Scheibe, E., Über einen verallgemeinerten affinen Zusammenhang, Math, Z, 57 (1952), 65-74.Google Scholar