Intrinsic characteristic classes of a local Lie group

Abstract

For a local Lie group M we define cohomology classes [is an obstruction to globalizability and give an example where [w1] = 0. We also show that [w3] coincides with Godbillon-Vey class in a particular case. These classes are secondary as they emerge when curvature vanishes.

The Infinitesimalization and Reconstruction of Locally Homogeneous Manifolds

“…Moreover, P = G 0 × {m 0 } and the development D :M → G 0 /H 0 is seen to be a diffeomorphism. This establishes (1).…”

“…Proof . The claim (1) is just a special case of Proposition 5.1 (1). Conclusion (2) is a special case of Proposition 6.5.…”

“…Of course, in the case that the embedding Γ ⊂ G 0 in (3) is normal, M is a global Lie group. The problem of globalizing a local Lie group structure is also considered in [1] (under the additional assumption that the connection∇ comes from a global parallelism on M ). In particular, it is shown that a necessary condition for globalizability is the vanishing of the class [w] ∈ H 1 (M ), where w is the closed one-form defined by w(U ) = trace(tor∇(U, · )), which is closed on account of Bianchi's second identity and the hypothesis curv ∇ = 0.…”

Riemannian geometry as a curved pre-homogeneous geometry

“…However we look at curvature, it must deform the "symmetric" object (G, M ) into a "lumpy" object. All the existing approaches to the concept of curvature (see, for instance, [14], [29], [6], [8]) circle around this fundamental idea and this note and [2] are no exceptions. It is commonly accepted today (at least in Riemannian geometry, see [15] and the recent work [8] for parabolic geometries) that this lumpy object is a principal bundle P → N with structure group H ≃ H p , dim N = dim M, together with some extra structure on P → N , like a torsionfree connection.…”

“…This note is the continuation of [24], [3], [2] and its main purpose is to carry out the program outlined in the introduction of [2] in the case of Riemannian and affine structures. So we will start by recalling this program in more technical detail than [2].…”