The Ambrose Singer Theorem
7 Pages
English
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The Ambrose Singer Theorem

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7 Pages
English

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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


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