Riemannian Foliation

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.