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AuthorMolino, Pierre. author
TitleRiemannian Foliations [electronic resource] / by Pierre Molino
ImprintBoston, MA : Birkhรคuser Boston, 1988
Connect tohttp://dx.doi.org/10.1007/978-1-4684-8670-4
Descript XII, 344 p. online resource

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 parยญ 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 [transverยญ 1--------1- - - -- sally], the leaves of such a 1--------1 - - - - -. stacking are the points of a 1--------1--- ----. quotient manifold W of diยญ L..... -' _ mension q. -----̃) W M Actually, this image corresponds to an elementary type of foliaยญ 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


CONTENT

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


Mathematics Geometry Differential geometry Mathematics Geometry Differential Geometry



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