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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Local holonomy groups of induced connections


Author: Mu Chou Liu
Journal: Proc. Amer. Math. Soc. 42 (1974), 272-278
MSC: Primary 53C05
MathSciNet review: 0331266
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Abstract: There are two naturally induced connections on the tangent bundle, the so called Jacobi connection and the Sasaki connection. By using the elementary theory of systems of linear differential equations, we completely determine the local holonomy group of these two induced connections, and find some relation to the local holonomy group of the manifold itself. There is an induced connection on the vector bundle of linear maps of the fibers. We also investigate the properties of the holonomy group of this bundle.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0331266-9
PII: S 0002-9939(1974)0331266-9
Keywords: Pseudo-Riemannian connection, Sasaki connection, holonomy group, parallel vector field, parallel section, affine fiber map, parallel translation, vector bundle, tangent bundle
Article copyright: © Copyright 1974 American Mathematical Society