We elaborate an unified geometric approach to classical mechanics, Riemann–Finsler spaces and gravity theories on Lie algebroids provided with nonlinear connection (N–connection) structure. There are investigated the conditions when the fundamental geometric objects like the anchor, metric and linear connection, almost sympletic and related almost complex structures may be canonically defined b...

In this note a functorial approach to the integration problem of an LAgroupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibered products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns out, that the fibered product of Lie algebroids along integrable morphisms is...

We apply the Atiyah–Singer index theorem and tensor products of elliptic complexes to cohomology transitive Lie algebroids. prove that Euler characteristic a representation algebroid $A$ over compact manifold $M$ vanishes unless $A=TM$, general Künneth formula. As applications, we give short proof vanishing result for principal bundle calculated using invariant differential forms, show certain ...

We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of s...

The purpose of this note is to give a “coordinate free” construction and prove the uniqueness of the vertex algebroid which gives rise to the chiral de Rham complex of [GMS]. In order to do that, we adapt the strategy of [BD] to the setting of vertex agebroids. To this end we show that the stack EVAX of exact vertex algebroids on X is a torsor under the stack in Picard groupoids ECAX of exact C...

In this paper we define Courant algebroids in a purely algebraic way and study their deformation theory by using two different but equivalent graded Poisson algebras of degree −2.