The Ambrose Singer Theorem

Niveau: Supérieur, Master, Bac+5
The Ambrose-Singer Theorem Codogni Giulio and Storch Matthias 7th January, 2011 Abstract In the following we will explain a proof of the Ambrose-Singer theorem. Given a principal bundle with structural group G, this theorem describes the Lie algebra of the holonomy group in term of the curvature form. We mainly follow the ideas of [KN63, Nom56]. 1 Prerequisites Definition 1.1 (Distribution). Let M be a manifold. A distribution E of dimension r on M assigns to each point p ?M an r dimensional subspace Ep of TpM . A distribution is called di?er- entiable, if every point p ?M has a neighborhood U and di?erentiable vector fields X1, . . . , Xr on U , such that X1q , . . . , X r q form a basis of basis of Eq for each q ? U . Such vector fields are called local base of the distribution E in U . A distribution is called involutive, we have [X,Y ]q ? Eq for all X,Y ? X (U) satisfying Xq, Yq ? Eq. Definition 1.2 (Integral Manifold). Let M be a manifold and E a distribution on M . A connected submanifold f : N ?? M is called integral manifold of a distribution E on M , if df(TpN) = Ef(p), where f is the embedding of N into M .

• reduction theorem


• group ?


• all horizontal


• spaces into horizontal


• ?x


• horizontal curve


• vector fields


• parameter group


• called involutive