The Journal of Geometric Mechanics (JGM)

Infinitesimally natural principal bundles

Pages: 199 - 220, Volume 8, Issue 2, June 2016      doi:10.3934/jgm.2016004

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Bas Janssens - Universiteit Utrecht, Budapestlaan 6, 3584 CD Utrecht, Netherlands (email)

Abstract: We extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. We classify the principal fibre bundles with this property. A version of the main result in this paper (theorem 4.4) can be found in Lecomte's work [12]. Our approach was developed independently, uses the language of Lie algebroids, and can be generalized in several directions.

Keywords:  Natural bundles, diffeomorphism group, integration of Lie algebroids, spin structures, Lie algebras of vector fields.
Mathematics Subject Classification:  Primary: 53C27, 57R50, 22E65, 57R22.

Received: June 2015;      Revised: April 2016;      Available Online: June 2016.