Lie algebroids and Cartan’s method of equivalence

Abstract

Abstract.Élie Cartan's general equivalence pr of Lie algebroids. The resulting formalism, bein allows for a full geometric interpretation of Carta reduction and prolongation. We show how to co (Cartan algebroids) for objects of finite-type, a directly as 'infinitesimal symmetries deformed b Details are developed for transitive structure include intransitive structures (intransitive symm illustrations include subriemannian contact stru try.c'est la dissymétrie qui crée le phénomène

Intrinsic characteristic classes of a local Lie group

“…Therefore, curvature is a measure of how much the geometry deviates from some local homogeneous model. This way of looking at curvature parallels the one proposed in the recent works [2], [3]. This paper is motivated by our attempt to do something globally interesting with the curvature in [22] for m = 0.…”

“…6 From Lie algebroid to Lie algebra Proposition 8 The following are equivalent: (3). By Lemma 5, j 1 ϑ = Γ(ϑ) for any ϑ ∈ X ε (M ) and therefore…”

“…On the other hand, the splitting (12) is conceptually different as we will see in Sections 5, 6. The pair ( Γ, Γ) is a dual pair of affine connections according to [2], [3]. Clearly, T i jk = 0 implies R 2 (ε) = R 1 (ε) = 0, but R 2 (ε) need not vanish in general as we will see below.…”

“…The pair ( Γ, Γ) is a dual pair of affine connections as defined in [2], [3]. In Section 4 we associate two curvatures R 1 (ε), R 2 (ε) to ε and compute them explicitly in coordinates.…”

Integration of quadratic Lie algebroids to Riemannian Cartan–Lie groupoids

“…Here we use the notations from [16] for the Lie algebroid connections τ ∇ and α ∇, induced by the ordinary connection ∇ on A (cf. [1,2] as well as [4,29,24] for the related gauge transformations). In the last condition above, R ∇ is the curvature and D the exterior covariant derivative of ∇, t is the A-torsion of α ∇, and ι ρ denotes the contraction with the T M-part of ρ ∈ Γ(A * ⊗ T M).…”