Integrability vs Non-Integrability of Distributions : Frobenius vs Chow

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Ngo, Khanh Quoc
Riemann Geometry
thesis / dissertation description
In this thesis, we explore the following question: what is the accessible set of a distribution H on a manifold M? In particular, when is the accessible set the entire manifold M and when does H give rise to a foliation of M? The first question is answered by a theorem of Chow, the second by a theorem of Frobenius. The intermediate case - when the accessible sets are of dimension between the dimensions of H and that of M- is answered be Sussmann's Orbit theorem. After introducing the necessary concepts, we prove Chow's and Frobenius Theorem ( the former one in a special case) and describe some of their applications