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Foliations admitting transverse systems of differential equations. (English) Zbl 0649.57027

Let M be a connected smooth manifold and \(J^ k({\mathbb{R}},M)\) be the bundle of k-jets of smooth mappings from \({\mathbb{R}}\) into M. A subbundle E of \(J^ k({\mathbb{R}},M)\) is called a system of differential equations of order k on the manifold M. The main aim of the paper is the study of properties of foliations F on M which admit a system of differential equations of order k, E, compatible (“foliated”) with F. This, following the author, supposes that there exists a subbundle Q supplementary to the tangent bundle of F, such that the set \(J^ k({\mathbb{R}},Q)\cap E\) is a foliated subbundle of the bundle \(J^ k({\mathbb{R}},Q)\) of k-jets of mappings f from \({\mathbb{R}}\) into M tangent to Q.
The author supposes, moreover, that the solutions of the system E compatible with F satisfy some conditions of uniqueness and smooth dependence on the initial conditions, as well as some conditions of transitivity and completeness of solutions of E transverse to F.
For a foliation F which admits a compatible system E satisfying the above conditions, the author proves, e.g., the following results: (1) The leaves of the foliation F have a common universal covering space. (2) If F is defined by a submersion \(h: M\to N\), then \(h: M\to N\) is a locally trivial fibre bundle.
The paper is interesting because the ideas developed in it lead a to unified study of Riemannian, transversally affine, transversally homogeneous conformal and \(\nabla\)-G foliations, among others.
Reviewer: E.Outerelo

MSC:

57R30 Foliations in differential topology; geometric theory
53C12 Foliations (differential geometric aspects)
58A30 Vector distributions (subbundles of the tangent bundles)
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References:

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