Geometric structures as deformed infinitesimal symmetries

Abstract

Abstract. A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie algebroid structure. The curvature of this connection vanishes precisely when the structure is locally symmetric.This model generalizes Cartan geometries, a substantial class, to the intransitive case. Simple examples are surveyed and corresponding local obstructions to symmetry are identified. These examples include foliations, Riemannian structures, infinitesimal G-structures, symplectic and Poisson structures.

Lie Algebroid Invariants for Subgeometry

“…Here so(T M ) is the so(n)-bundle of skew-symmetric endomorphisms of the tangent bundle T M (the kernel of the anchor of so[T M ]) and the second isomorphism is obtained with the help of the Levi-Cevita connection. In this model the connection ∇ coincides with the canonical Cartan connection on so[T M ], in the sense of [2], constructed for arbitrary Riemannian manifolds in [3]. Readers familiar with tractor bundles will recognise the model on the right of (1.3) as the adjoint tractor bundle associated with M [12].…”

“…1. By construction, the flat connection ∇ drops to a connection on A(f ) k that is actually a Cartan connection (in the sense of [2]). In particular, the space s of ∇-parallel sections of A(f ) k is a Lie subalgebra of all sections, and acts infinitesimally on the submanifold Σ.…”

Cartan Connections on Lie Groupoids and their Integrability

“…We begin by recalling that every Lie algebroid equipped with a flat Cartan connection ∇ is an action algebroid, if M is simply-connected [3] (for generalizations, see [5]): Proof of Proposition 5.2. Applying the lemma to the Lie algebra g of G, we obtain a Lie algebra g 0 and a morphism of Lie algebroids ω : g → g 0 , such that ω restricted to any fibre g| m is an isomorphism onto g 0 .…”

“…The present article provides some detail missing from our work published on related matters [3,4,5,6] and is more technical than those works. Applications of the Lie groupoid approach to Cartan connections will be given elsewhere.…”

“…The Lie algebroid of J 1 G can be identified with J 1 g, where g denotes the Lie algebroid of G (see Section 3.1 below). The canonical exact sequence,0 −→ T * M ⊗ g −→ J 1 g −→ g −→ 0 (3.1)may be regarded as the derivative of a natural sequence of groupoid morphisms,Aut(T M , g) → J 1 G → G.HereAut(T M , g) is a certain open neighborhood of the zero-section of T * M ⊗ g, defined in Section 3.4 below; if G = M × M , then Aut(T M , g) = Aut(T M ).Corresponding to (3.1) is an exact sequence of section spaces which splits canonically, leading to an identification,Γ J 1 g ∼ = Γ(g) ⊕ Γ(T * M ⊗ g).Under this identification, the Lie bracket on Γ(J 1 g) is a semidirect product[3].In this section we establish the global analogue of this result, namely a semidirect product structure,B J 1 G ∼ = B(G) × B(Aut(T M , g)),where B( · ) denotes the group of global bisections.Just as a choice of Cartan connection ∇ on g determines an identificationJ 1 g ∼ = g ⊕ (T * M ⊗ g)and an associated semi-direct product structure for J 1 g, so a choice of Cartan connection on G determines a semi-direct product structure for J 1 G. While we shall provide a direct demonstration of this fact, the reader may like to interpret the existence of the semi-direct product…”