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.
Inhaltsverzeichnis
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.
