Intersection between the geometry of generalized Lie algebroids and some aspects of interior and exterior differential systems

TL;DR

This paper develops an exterior differential calculus framework for generalized Lie algebroids, establishing fundamental theorems like Maurer-Cartan, Cartan, and Bianchi identities, and characterizes involutivity of interior differential systems.

Contribution

It introduces a new exterior calculus approach for generalized Lie algebroids and extends classical involutivity theorems to this broader context.

Findings

Established Maurer-Cartan type theorem for generalized Lie algebroids.

Derived Cartan and Bianchi identities within this framework.

Characterized involutivity of interior differential systems in generalized Lie algebroids.

Abstract

An exterior differential calculus in the general framework of generalized Lie algebroids is presented. A theorem of Maurer-Cartan type is obtained. All results with details proofs are presented and a new point of view over exterior differential calculus for Lie algebroids is obtained. Using the theory of linear connections of Ehresmann type presented in the firstt reference, the identities of Cartan and Bianchi type are presented. Supposing that any vector subbundle of the pull-back Lie algebroid of a generalized Lie algebroid is interior differential system (IDS) for that generalized Lie algebroid, then the involutivity of the IDS in a theorem of Frobenius type is characterized. Extending the classical notion of exterior differential system (EDS) to generalized Lie algebroids, then the involutivity of an IDS in a theorem of Cartan type is characterized.