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Liftings of Some Types of Tensor Fields and Connections to Tangent Bundles of pr-Velocities

Published online by Cambridge University Press:  22 January 2016

Akihiko Morimoto
Akihiko Morimoto*
Affiliation:
Nagoya University
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In the previous paper [6], we studied the liftings of tensor fields to tangent bundles of higher order. The purpose of the present paper is to generalize the results of [6] to the tangent bundles of pr-velocities in a manifold M— notions due to C. Ehresmann [1] (see also [2]). In §1, we explain the pr-velocities in a manifold and define the (Λ)-lifting of differentiable functions for any multi-index λ -(λ1, λ2,…,λp) of non-negative integers λi satisfying ΣΛi≤r. In § 2, we construct ‹Λ›-lifts of any vector fields and ‹Λ›-lifts of 1-forms. The ‹Λ›-lift is a little bit different from the ‹Λ›-lift of vector fields in [6].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 40 , December 1970 , pp. 13 - 31
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1970

References

[1] Ehresmann, C., Les prolongements d’une variete differentiable I. Calcul des jets, prolongement principal, C.R. Acad. Sci. Paris, 233(1951), 598–600.Google Scholar
[2] Ehresmann, C., Les prolongements d’une variete differentiable II, L’espace des jets d’ordre r de Vn dans Vm , ibid., 777–779.Google Scholar
[3] Helgason, S., Differential geometry and symmetric spaces, Acad. Press, 1962.Google Scholar
[4] Morimoto, A., Prolongations of G-structures to tangent bundles of higher order, Nagoya Math. J. 38(1970), 153–179.CrossRefGoogle Scholar
[5] Morimoto, A., Prolongations of connections to tangential fibre bundles of higher order, to appear.Google Scholar
[6] Morimoto, A., Liftings of tensor fields and connections to tangent bundles of higher order, to appear.Google Scholar