The authors have Introduced In the last few years a
construction of characteristic classes for foliated bundles which
provides among other things a construction of characteristic
Invariants of foliations. The purpose of these lectures Is to present this
construction, and to Interprete and compute these new Invariants In
various geometric contexts.
The basic concept In this theory Is a foliated bundle. This
Is a principal bundle P with a foliation on the base space M, and
a partial connection on P which Is only defined along the leaves
of the foliation on M, and which has zero curvature (definition
2.1). For the trivial foliation of M by points this concept
reduces to an ordinary principal bundle P and then our construction
of characteristic classes reduces to the ordinary Chern-Weil
construction. Our construction was Inspired by the work of Chern-
Slmons [CS 1]. For the trivial foliation of M consisting of one
single leaf the concept of a foliated bundle reduces to a flat
bundle. Our work on flat bundles (summarized in [KT 1]) was one
of our motivations for the introduction of the concept of a foliated
bundle in [KT 2,31. This concept allows the simultaneous discussion
of ordinary bundles, bundles with an infinitesimal or global group
action, flat bundles and normal bundles of foliations. For a
foliation, the canonical foliated bundle structure on the frames
normal to the foliation gives rise to characteristic Invariants
attached to the foliation. For this situation, the construction In
[KT 4-7] is one of the various Independently discovered approaches to
characteristic Invariants by Bernsteln-Rosenfeld [BR 1] [BR 2],
Bott-Haefllger [B 3] [H 5] [BH], Godblllon-Vey [GV], Malgrange
(not published) and the authors.