Journal of Generalized Lie Theory and Applications

Generalized Lie Algebroids and Connections over Pair of Diffeomorphic Base Manifolds

Constantin M. Arcus¸

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Abstract

Extending the definition of Lie algebroid from one base manifold to a pair of diffeomorphic base manifolds, we obtain the generalized Lie algebroid. When the diffeomorphisms used are identities, then we obtain the definition of Lie algebroid. We extend the concept of tangent bundle, and the Lie algebroid generalized tangent bundle is obtained. In the particular case of Lie algebroids, a similar Lie algebroid with the prolongation Lie algebroid is obtained. A new point of view over (linear) connections theory of Ehresmann type on a fiber bundle is presented. These connections are characterized by a horizontal distribution of the Lie algebroid generalized tangent bundle. Some basic properties of these generalized connections are investigated. Special attention to the class of linear connections is paid. The recently studied Lie algebroids connections can be recovered as special cases within this more general framework. In particular, all results are similar with the classical results. Formulas of Ricci and Bianchi type and linear . connections of Levi-Civita type are presented.

Article information

Source
J. Gen. Lie Theory Appl., Volume 7 (2013), 32 pages.

Dates
First available in Project Euclid: 23 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.jglta/1437656919

Digital Object Identifier
doi:10.4303/jglta/G111202

Mathematical Reviews number (MathSciNet)
MR3047637

Zentralblatt MATH identifier
1268.58006

Subjects
Primary: 00A69: General applied mathematics {For physics, see 00A79 and Sections 70 through 86} 58B34: Noncommutative geometry (à la Connes) 53B05: Linear and affine connections

Citation

M. Arcus¸, Constantin. Generalized Lie Algebroids and Connections over Pair of Diffeomorphic Base Manifolds. J. Gen. Lie Theory Appl. 7 (2013), 32 pages. doi:10.4303/jglta/G111202. https://projecteuclid.org/euclid.jglta/1437656919


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