Ebook: Foliations on Riemannian Manifolds

A first approximation to the idea of a foliation is a dynamical system, and the resulting decomposition of a domain by its trajectories. This is an idea that dates back to the beginning of the theory of differential equations, i.e. the seventeenth century. Towards the end of the nineteenth century, Poincare developed methods for the study of global, qualitative properties of solutions of dynamical systems in situations where explicit solution methods had failed: He discovered that the study of the geometry of the space of trajectories of a dynamical system reveals complex phenomena. He emphasized the qualitative nature of these phenomena, thereby giving strong impetus to topological methods. A second approximation is the idea of a foliation as a decomposition of a manifold into submanifolds, all being of the same dimension. Here the presence of singular submanifolds, corresponding to the singularities in the case of a dynamical system, is excluded. This is the case we treat in this text, but it is by no means a comprehensive analysis. On the contrary, many situations in mathematical physics most definitely require singular foliations for a proper modeling. The global study of foliations in the spirit of Poincare was begun only in the 1940's, by Ehresmann and Reeb.

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This book introduces the reader to the theory of foliations, and explores some of the interactions of foliations with the Riemannian geometry of the ambient manifold. The author begins with motivations for the study of the subject and proceeds to discuss the instructive special case of transversally oriented foliations of codimension one. The second part of the book covers such topics as foliations by level hypersurfaces, infinitesimal automorphisms and basic forms, flows, Lie foliations, and twisted duality.

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This book introduces the reader to the theory of foliations, and explores some of the interactions of foliations with the Riemannian geometry of the ambient manifold. The author begins with motivations for the study of the subject and proceeds to discuss the instructive special case of transversally oriented foliations of codimension one. The second part of the book covers such topics as foliations by level hypersurfaces, infinitesimal automorphisms and basic forms, flows, Lie foliations, and twisted duality.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-7
Integrable Forms....Pages 8-23
Foliations....Pages 24-34
Flat Bundles and Holonomy....Pages 35-46
Riemannian and Totally Geodesic Foliations....Pages 47-61
Second Fundamental Form and Mean Curvature....Pages 62-73
Codimension One Foliations....Pages 74-103
Foliations by Level Hypersurfaces....Pages 104-116
Infinitesimal Automorphisms and Basic Forms....Pages 117-131
Flows....Pages 132-142
Lie Foliations....Pages 143-148
Twisted Duality....Pages 149-163
A Comparison Theorem....Pages 164-168
Back Matter....Pages 169-247

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This book introduces the reader to the theory of foliations, and explores some of the interactions of foliations with the Riemannian geometry of the ambient manifold. The author begins with motivations for the study of the subject and proceeds to discuss the instructive special case of transversally oriented foliations of codimension one. The second part of the book covers such topics as foliations by level hypersurfaces, infinitesimal automorphisms and basic forms, flows, Lie foliations, and twisted duality.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-7
Integrable Forms....Pages 8-23
Foliations....Pages 24-34
Flat Bundles and Holonomy....Pages 35-46
Riemannian and Totally Geodesic Foliations....Pages 47-61
Second Fundamental Form and Mean Curvature....Pages 62-73
Codimension One Foliations....Pages 74-103
Foliations by Level Hypersurfaces....Pages 104-116
Infinitesimal Automorphisms and Basic Forms....Pages 117-131
Flows....Pages 132-142
Lie Foliations....Pages 143-148
Twisted Duality....Pages 149-163
A Comparison Theorem....Pages 164-168
Back Matter....Pages 169-247
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