Ebook: Riemannian Foliations

Foliation theory has its origins in the global analysis of solutions of ordinary differential equations: on an n-dimensional manifold M, an [autonomous] differential equation is defined by a vector field X ; if this vector field has no singularities, then its trajectories form a par­ tition of M into curves, i.e. a foliation of codimension n - 1. More generally, a foliation F of codimension q on M corresponds to a partition of M into immersed submanifolds [the leaves] of dimension ,--------,- - . - -- p = n - q. The first global image that comes to mind is 1--------;- - - - - - that of a stack of "plaques". 1---------;- - - - - - Viewed laterally [transver­ 1--------1- - - -- sally], the leaves of such a 1--------1 - - - - -. stacking are the points of a 1--------1--- ----. quotient manifold W of di­ L..... -' _ mension q. -----~) W M Actually, this image corresponds to an elementary type of folia­ tion, that one says is "simple". For an arbitrary foliation, it is only l- u L ally [on a "simpIe" open set U] that the foliation appears as a stack of plaques and admits a local quotient manifold. Globally, a leaf L may - - return and cut a simple open set U in several plaques, sometimes even an infinite number of plaques.

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Content:
Front Matter....Pages i-xii
Elements of Foliation Theory....Pages 1-31
Transverse Geometry....Pages 33-67
Basic Properties of Riemannian Foliations....Pages 69-101
Transversally Parallelizable Foliations....Pages 103-145
The Structure of Riemannian Foliations....Pages 147-183
Singular Riemannian Foliations....Pages 185-216
Back Matter....Pages 217-343

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Content:
Front Matter....Pages i-xii
Elements of Foliation Theory....Pages 1-31
Transverse Geometry....Pages 33-67
Basic Properties of Riemannian Foliations....Pages 69-101
Transversally Parallelizable Foliations....Pages 103-145
The Structure of Riemannian Foliations....Pages 147-183
Singular Riemannian Foliations....Pages 185-216
Back Matter....Pages 217-343
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