CHAPTER 1

Introduction

The qualitative study of foliations is well developed for codimension one. As

for higher codimension the investigation in general becomes extremely difficult.

However there are some special cases for which detailed research is possible. One

of them, which we are going to study in this paper, is a foliation with a transverse

geometric structure.

There are various geometric structures with different characters, some easy to

deal with and others hard to analyze. The easiest of all is perhaps a Lie G structure.

Foliations with transverse Lie G structures exhibit nice geometric properties. For

example they have no nontrivial leaf holonomy, and admit transversely invariant

Riemanniann metrics. Lie G flows on closed 3-manifolds are already classified by

Y. Carriere [13], [10], [11]. Also Lie G foliations for G the nonabelian simply

connected 2 dimensional Lie group are classified on closed 4 manifolds [29].

Turning to other geometric structures, W. P. Thurston [40] classified trans-

versely hyperbolic flows in dimension three and D. B. A. Epstein [16] generalized

the classification to general dimension. Also flows with transverse Euclidean simi-

larity structures are studied by E. Ghys [21], and especially a complete classifica-

tion is obtained for dimension 3. See also T. Nishimori [33] and M. Brunella [6]

for related topics. Quite recently T. Asuke [1, 2] classified flows with transverse

similarity structures in general dimension, but with an additional condition.

However for general affine structures, almost nothing is known so far. This is

due to the difficulties arising from the higher discompacity of the Lie group of the

affine transformations, where the concept of discompacity is defined in [12].

The purpose of this paper is a detailed study of transversely affine flows on

3-manifolds. However the difficulty mentioned above prevents us to carry out a

general treatment. We have to confine ourselves to special kinds of flows.

A dimension one foliation (pona 3-manifold M is called an affine flow if it is

transversely modelled smoothly on the geometry

(G^4+,M2),

where

GA+

is the Lie

group of the orientation preserving affine transformations of

R2.

Throughout this

paper we assume for simplicity that the manifold M, and therefore the foliation (/?,

is oriented.

Given a covering map q : T V — M and an affine flow p on M, the lift of p by

q, also an affine flow, denoted by q*ip, is defined in a natural way.

Let pi be an affine flow on a 3-manifold Mi (i = 1,2). The flow p\ is called

isomorphic to p2 if there is a diffeomorphism h : M\ — Mi such that h*cp2 = pi

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Received by the editor June 23, 1999; and in revised form March 4, 2002.

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