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Connexions and Prolongations

Published online by Cambridge University Press:  20 November 2018

Leif-Norman Patterson
Leif-Norman Patterson*
Affiliation:
University of Tromsø, Tromsø Norway
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Computation of the velocity of a given motion depends on measurement of nearby position changes only. Computation of acceleration, on the other hand, depends on measurement of nearby changes in velocity. But since velocity vectors are attached to positions so that even nearby ones are not a priori comparable, acceleration is not computable until a rule for comparison of vectors along a curve is given. Such a rule-parallel translation or linear connexion - exists automatically in Euclidean spaces. For motions in more general manifolds, for example (semi-) Riemannian ones, parallel translation is a less obvious consequence of the metric properties.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 4 , 01 August 1975 , pp. 766 - 791
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Dombrowski, P., On the geometry of the tangent bundle, J. Reine Angew. Math. 210 (1962), 73–88.Google Scholar
2. Greub, W., Halperin, S., and Vanstone, R., Connections, curvature and cohomology, Vol. 2, Ch. 7, (Academic Press, New York, 1973).Google Scholar
3. Ishihara, S. and Yano, K., Tangent and cotangent bundles, (Marcel Dekker, Inc., New York, 1973).Google Scholar
4. Kobayashi, S., Theory of connections, Ann. Mat. Pura Appl. 43 (1957), 119–194.Google Scholar
5. Kobayashi, S., Canonical forms on frame bundles of higher order contact, Proc. of the Symposia in Pure Mathematics III, Amer. Math. Soc. (1961).Google Scholar
6. Libermann, P., Calcul tensoriel et connexions d'ordre supérieur, 3° Côloquio Brasileiro de Matematica, Fortaleza, Brazil (1961).Google Scholar
7. Pohl, W., Connexions in differential geometry of higher order, Applied Mathematics and Statistics Laboratory, Stanford University (1963).Google Scholar
8. Weil, A., Théorie des points proches sur les variétés differentiables, Colloques Intern? CNRS, Strassbourg (1953), pp. 111–117.Google Scholar