Generalized Lie Algebroids and Connections over Pair of Diffeomorphic Base Manifolds

Constantin M Arcuş

doi:10.4303/jglta/g111202

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.

Highlights

If C is a category, we denote |C| the class of objects
Let Liealg, Mod, Man, B, and Bv be the category of Lie algebras, modules, manifolds, fiber bundles, and vector bundles, respectively
The theory of connections constitutes one of the most important chapter of differential geometry, which has been explored in the literature

Summary

Introduction

If C is a category, we denote |C| the class of objects. For any A, B ∈ |C|, we denote C(A, B) the set of morphisms of A source and B target, and IsoC(A, B) the set of C-isomorphisms of A source and B target. Langerock and Cantrijn [2] proposed a general notion of connection on a fiber bundle (E, π, M ) as being a smooth linear bundle map Γ ∈ Man(π∗(F ), T E) so that the diagram is commutative: π∗(F ) Γ / T E. where (F, ν, M ) is an arbitrary vector bundle and (ρ, IdM ) is a vector bundle morphism of (F, ν, M ) source and (T M, τM , M ) target. Using our linear connections theory, we succeed to extend the set Cov0(E,π,M) of Yang-Mills theory, because using all generalized Lie algebroid structures for the tangent bundle (T M, τM , M ), we obtain all possible linear connections for the vector bundle (E, π, M ). Using our theory of linear connections, the formulas of Ricci and Bianchi type and linear connections of Levi-Civita type are presented

Preliminaries

The Lie algebroid generalized tangent bundle

The Lie algebroid generalized tangent bundle of dual vector bundle