Fiber Bundles and Foliations

César Camacho and Alcides Lins Neto

Chapter V in Geometric Theory of Foliations, 1985, pp 87-113 from Springer

Abstract: Abstract The notion of holonomy of a leaf introduced in the previous chapter is essentially of local character. It is defined by a group of germs of diffeomorphisms of a transverse section to a leaf, with a fixed point. In certain circumstances, however, it is possible to associate to the foliation a group of diffeomorphisms of a global transverse section, containing in a certain well-defined sense the holonomy of each leaf. This is the case of foliations whose leaves meet transversely all the fibers of a fiber bundle E. The importance of these foliations is in the fact that they are characterized by their holonomy, in this case given by a representation ϕ : π 1 (B) ➞ Diff (F) of the fundamental group of the base of E to the group of diffeomorphisms of the fiber of E. In this manner properties of ϕ translate to properties of the foliation. For example, the action ϕ has exceptional minimal sets if and only if the same occurs for the foliation. Sacksteder’s example, of a C ∞ codimension one foliation with an exceptional minimal set, is a typical case of what we will see in this chapter.

Keywords: Vector Bundle; Fundamental Group; Fiber Bundle; Normal Bundle; Geometric Theory (search for similar items in EconPapers)
Date: 1985
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DOI: 10.1007/978-1-4612-5292-4_6

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