Geometry of Distributions on a Manifold -- Structural and Transversal Geometry of Foliations -- Foliations on Semi-Riemannian Manifolds -- Parallel Foliations -- Foliations Induced by Geometric Structures -- A Gauge Theory on a Vector Bundle.

This self-contained book starts with the basic material on distributions and foliations. It then gradually introduces and builds the tools needed for studying the geometry of foliated manifolds. The main theme of the book is to investigate the interrelations between foliations of a manifold on the one hand, and the many geometric structures that the manifold may admit on the other hand. Among these structures are: affine, Riemannian, semi-Riemannian, Finsler, symplectic, complex and contact structures. Using these structures, the book presents interesting classes of foliations whose geometry is very rich and promising. These include the classes of: Riemannian, totally geodesic, totally umbilical, minimal, parallel non-degenerate, parallel totally - null, parallel partially - null, symmetric, transversally symmetric, Lagrange, totally real and Legendre foliations. Some of these classes appear for the first time in the literature in book form. Finally, the vertical foliation of a vector bundle is used to develop a gauge theory on the total space of a vector bundle.