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Connexions and Prolongations

  • Leif-Norman Patterson (a1)

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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.

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References

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1. Dombrowski, P., On the geometry of the tangent bundle, J. Reine Angew. Math. 210 (1962), 7388.
2. Greub, W., Halperin, S., and Vanstone, R., Connections, curvature and cohomology, Vol. 2, Ch. 7, (Academic Press, New York, 1973).
3. Ishihara, S. and Yano, K., Tangent and cotangent bundles, (Marcel Dekker, Inc., New York, 1973).
4. Kobayashi, S., Theory of connections, Ann. Mat. Pura Appl. 43 (1957), 119194.
5. Kobayashi, S., Canonical forms on frame bundles of higher order contact, Proc. of the Symposia in Pure Mathematics III, Amer. Math. Soc. (1961).
6. Libermann, P., Calcul tensoriel et connexions d'ordre supérieur, 3° Côloquio Brasileiro de Matematica, Fortaleza, Brazil (1961).
7. Pohl, W., Connexions in differential geometry of higher order, Applied Mathematics and Statistics Laboratory, Stanford University (1963).
8. Weil, A., Théorie des points proches sur les variétés differentiables, Colloques Intern? CNRS, Strassbourg (1953), pp. 111117.
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Connexions and Prolongations

  • Leif-Norman Patterson (a1)

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