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Affine flows on 3-manifolds. (English) Zbl 1026.57022

Mem. Am. Math. Soc. 771, 94 p. (2003).
The foliations of dimension one with transverse Lie \(G\)-structures on \(3\)-manifolds, which possess nice geometric properties, have been previously studied by several authors. The classification of such foliations is, in general, a difficult problem which is solved only in some special cases (see, for example, W. P. Thurston [The geometry and topology of three manifolds, Preprint, Princeton University, chapter 4], for classification of transversely hyperbolic flows on three manifolds and E. Ghys [Mem. Soc. Math. Fr. 46, 123-150 (1991; Zbl 0761.57016)], for the study of flows with transverse Euclidian similarity structures).
The given paper is devoted to the study of foliations of dimension one on closed \(3\)-manifolds which are transversely modeled on the geometry \((GA^+, {\mathbb R}^2)\), where \(GA^+\) is the Lie group of the orientation preserving affine transformations of \({\mathbb R}^2\). The results obtained by the author in this paper are new and of great interest.
For a given affine flow \(\varphi\) on \(M\), let \(\xi\colon \widetilde{M}\to {\mathbb R}^2\) be the developing map and \(\theta\colon \Pi\to GA^+\) the holonomy homomorphism associated with \(\varphi\), where \(\widetilde{M}\) is the universal covering space of \(M\) and \(\Pi\) is the fundamental group of \(M\). Then the holonomy group \(\Gamma =\theta (\Pi)\) satisfies a certain equivalence relation.
In Chapter 2 of this paper, the author studies complete affine flows on closed \(3\)-manifolds. Here an affine flow \(\varphi\) is said to be complete if \(\xi\colon \widetilde{M}\to {\mathbb R}^2\) is a trivial \({\mathbb R}\)-bundle over \({\mathbb R}^2\). Concrete examples of complete affine flows are given by Example 1.4. The main result in Chapter 2 by the author is that any complete flow on a closed oriented \(3\)-manifold is virtually isomorphic to either one given in Example 1.4. Note that the completing condition is not given in terms of the dynamics or the geometry of the flow and the author’s goal is to replace it by a dynamically natural condition. Therefore, the author next considers the classes of affine flows whose holonomy groups satisfy a strong condition and no additional conditions, like the completing one, are imposed.
The main result in Chapter 4 is the classification of affine flows on \(3\)-oriented closed manifolds with the holonomy group contained in the Lie group \(SL=SL(2,{\mathbb R})\). Some kind of foliations, called luxuriant, appear in a natural way, associated with the \(SL\)-flows. Note that the concept of luxuriant foliation is a generalization of the weak unstable foliations of Anosov flows. In general, a transversely oriented codimention one foliation \(\mathcal F\) on a closed \(3\)-manifold \(M\) is called luxuriant if it is defined by a \(C^r(r\geq 2)\) \(1\)-form \(\omega\) so that \(d\omega = \omega _1\wedge \omega\) for some \(1\)-form \(\omega _1\) and \(d\omega \) vanishes nowhere. The oriented one-dimensional foliation \(\varphi\) defined by \(\ker (d\omega)\) is called the horizontal flow associated to the \(1\)-form \(\omega\). In Chapter 3, the author establishes the fundamental properties of luxuriant foliations and the horizontal flows associated with them. In particular, any annular leaf of \({\mathcal F}\) has nontrivial holonomy. Note also that luxuriant foliations are possible candidates to produce the new (unknown) examples of minimal flows. In Chapter 5 of the paper, the author gives the classification of \(SA\)-flows \(\varphi\) on \(3\)-manifolds with the homotopy lifting property (HLP), where the holonomy group \(\Gamma\) of \(\varphi\) is contained in the Lie group \(SA=SL(2,{\mathbb R})\ltimes {\mathbb R}^2\). This result is related to the similar results in Sections 2 and 4. In many cases the flow \(\varphi\) is accompanied with a codimensional one foliation, whose leaves are affine subsurfaces, defined by the author.

MSC:

57R30 Foliations in differential topology; geometric theory
37C10 Dynamics induced by flows and semiflows
57R25 Vector fields, frame fields in differential topology
53C12 Foliations (differential geometric aspects)
57M50 General geometric structures on low-dimensional manifolds

Citations:

Zbl 0761.57016
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