By Philippe Tondeur (auth.)

A first approximation to the belief of a foliation is a dynamical approach, and the ensuing decomposition of a website by means of its trajectories. this is often an concept that dates again to the start of the speculation of differential equations, i.e. the 17th century. in the direction of the top of the 19th century, Poincare constructed equipment for the examine of world, qualitative houses of options of dynamical structures in events the place specific answer equipment had failed: He came across that the research of the geometry of the gap of trajectories of a dynamical approach finds complicated phenomena. He emphasised the qualitative nature of those phenomena, thereby giving powerful impetus to topological equipment. A moment approximation is the belief of a foliation as a decomposition of a manifold into submanifolds, all being of a similar size. right here the presence of singular submanifolds, akin to the singularities when it comes to a dynamical procedure, is excluded. this can be the case we deal with during this textual content, however it is in no way a entire research. to the contrary, many occasions in mathematical physics most likely require singular foliations for a formal modeling. the worldwide research of foliations within the spirit of Poincare was once started merely within the 1940's, by way of Ehresmann and Reeb.

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If H is a closed connected subgroup, the coset foliation is defined by the submersion G ~ G/H. Let r be a discrete subgroup of G with orbit space r \ G (left action). The foliation of G by H passes to a foliation of r \ G. are locally homogeneous foliations. These A typical example of such a foliation for G = PSL(2,~) was discussed in Chapter 2. Pullback of foliations. Let f: M~ N be a smooth map, transverse to a foliation 1 on N. This means that f*(TxM) + Lf(x) x E M. = T1(x)N for each An example is a submersion and the foliation by points on N.

This is in contrast to the Chern-Weil construction, where differential forms are constructed from the curvature form alone. This discussion hints at the generalized Godbillon-Vey construction, using more general products of connection and curvature forms. See [BO 2] [HA 4] [KT 3] for discussions of this topic. A final remark on the topic of integrable i-forms. condition

Thus we obtain on M a foliation f *1 which has the same codimension as 1 on N. 34 In the next chapter we turn to a particularly interesting class of foliations. CHAPTER 4 FLAT BUNDLES AND HOLONOMY We begin by discussing a construction of flat, bundles. be a diffeomorphism of a smooth manifold F. the inverse images of the projection (t,y)n = (t+n,hn(y)) ~ x F Let h: F ~ F The product foliation defined by ~ F is invariant under the action of hll, n E 1. This means that the quotient ~ carries a 1-dimensional foliation transverse to the fibers of ~ x IF x IF ~ ~/l ~ S1.