We extend the formality theorem of M. Kontsevich from deformations of the structure sheaf on a manifold to deformations of gerbes.

In the present paper, two optimal control problems are studied using Lie geometric methods and applying Pontryagin Maximum Principle at level of a new working space, called algebroid. It is proved that framework algebroid more suitable than cotangent bundle in order to find solutions some driftless affine systems with holonomic distributions. Finally, an economic application given.

We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibred over IR. It is argued that this is the framework which one needs for coming to a time-dependent generalization of the theory of Lagrangian systems on Lie algebroids. An extensive discussion is given of a way one can think of forms acting on sections of the affine bundle. It is further shown th...

In the first section we discuss Morita invariance of differentiable/algebroid cohomology. In the second section we extend the Van Est isomorphism to groupoids. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and conjectured in degree 2 by Weinstein-Xu [50]). As a second application we extend Van Est’s argument for the integra...

A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but i...

We extend R. Fernandes’ construction of the secondary characteristic classes of a Lie algebroid to the case of a base-preserving morphism between two Lie algebroids. Like in the case of a Lie algebroid, the simplest characteristic class of our construction coincides with the modular class of the morphism. In [4] R. Fernandes has constructed a sequence of secondary characteristic classes of a Li...

We construct a new class of exact solutions describing spacetimes possessing Lie algebroid symmetry. They are described by generic off– diagaonal 5D metrics embedded in bosonic string gravity and possess nontrivial limits to the Einstein gravity. While we focus on nonholonomic vielbein transforms of the Schwarzschild metrics to 5D ansatz with solitonic backgrounds, much of the analysis continue...