Cover Image
Read Now

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 int...

Full description

Bibliographic Details
Main Author: Molino
Format: eBook
Language:English
Published: Boston, MA Birkhäuser Boston 1988, 1988
Edition:1st ed. 1988
Series:Progress in Mathematics
Subjects:
Differential Geometry
Geometry
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
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
  • 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
  • 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

Similar Items