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, --, - - - -- …
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- Full title :Riemannian foliations / Pierre Molino ; translated by Grant Cairns ; with appendices by G. Cairns [and others].
Summary
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
Read More ↓Edition Statements
- Publication Date:1988.
Publication Statements
- Publisher(s):Birkhäuser
- Place of Publication:Boston
- Language:eng
- Publication Date:1988.
The Physical Object
- Description:1 online resource (xii, 339 pages) : : illustrations
Edition Identifiers
- ISBN:9781468486704, 1468486705
- DOI:10.1007/978-1-4684-8670-4
- oclc:ocn679955604
- LC Classification:QA613.62
- Dewey Decimal:514/.72
Table of Contents
1 Elements of Foliation theory -- 1.1. Foliated atlases ; foliations -- 1.2. Distributions and foliations -- 1.3. The leaves of a foliation -- 1.4. Particular cases and elementary examples -- 1.5. The space of leaves and the saturated topology -- 1.6. Transverse submanifolds ; proper leaves and closed leaves -- 1.7. Leaf holonomy -- 1.8. Exercises -- 2 Transverse Geometry -- 2.1. Basic functions -- 2.2. Foliate vector fields and transverse fields -- 2.3. Basic forms -- 2.4. The transverse frame bundle -- 2.5. Transverse connections and G-structures -- 2.6. Foliated bundles and projectable connections -- 2.7. Transverse equivalence of foliations -- 2.8. Exercises -- 3 Basic Properties of Riemannian Foliations -- 3.1. Elements of Riemannian geometry -- 3.2. Riemannian foliations: bundle-like metrics -- 3.3. The Transverse Levi-Civita connection and the associated transverse parallelism -- 3.4. Properties of geodesics for bundle-like metrics -- 3.5. The case of compact manifolds : the universal covering of the leaves -- 3.6. Riemannian foliations with compact leaves and Satake manifolds -- 3.7. Riemannian foliations defined by suspension -- 3.8. Exercises -- 4 Transversally Paralleli…
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