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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ISBN-13
978-1-4684-8670-4 (9781468486704)
DOI
10.1007/978-1-4684-8670-4
Schweitzer Klassifikation
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 Parallelizable Foliations.- 4.1. The basic fibration.- 4.2. CompIete Lie foliations.- 4.3. The structure of transversally parallelizable foliations.- 4.4. The commuting sheaf C(M, F).- 4.5. Transversally complete foliations.- 4.6. The Atiyah sequence and developability.- 4.7. Exercises.- 5 The Structure of Riemannian Foliations.- 5.1. The lifted foliation.- 5.2. The structure of the leaf closures.- 5.3. The commuting sheaf and the second structure theorem.- 5.4. The orbits of the global transverse fields.- 5.5. Killing foliations.- 5.6. Riemannian foliations of codimension 1, 2 or 3.- 5.7. Exercises.- 6 Singular Riemannian Foliations.- 6.1. The notion of a singular Riemannian foliation.- 6.2. Stratification by the dimension of the leaves.- 6.3. The local decomposition theorem.- 6.4. The linearized foliation.- 6.5. The global geometry of SRFs.- 6.6. Exercises.- Appendix A Variations on Riemannian Flows.- Appendix B Basic Cohomology and Tautness of Riemannian Foliations.- Appendix C The Duality between Riemannian Foliations and Geodesible Foliations.- Appendix D Riemannian Foliations and Pseudogroups of Isometries.- Appendix E Riemannian Foliations: Examples and Problems.- References.
