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Variétés affines radiales de dimension 3. (Radiant affine \(3\)-manifolds). (French. English, French summaries) Zbl 0954.57003

A radiant affine structure on an \(n\)-manifold \(M\) is an \((\text{GL}(\mathbb{R}), \mathbb{R}^n)\)-structure for \(M\), where \(\text{GL}(n, \mathbb{R})\) is the general linear group of real \(n \times n\) matrices acting linearly on \(\mathbb{R}^n\). This is a particular case of \((G, X)\)-structure for a manifold, where \(G\) is a Lie group acting analytically on the \(C ^\omega\)-manifold \(X\). The terminology “radiant affine structure” comes from the existence of a particular non-singular flow on \(M\), namely the radial flow, which preserves the affine structure. This flow on \(M\) is generated by the vector field induced by the vector field \(\sum_{k=1}^{n} x^k \frac{\partial}{\partial x^k} \) on \(\mathbb{R}^n\), via the developing mapping \(\mathbb D: \widetilde{M} \rightarrow X\), where \(\widetilde{M}\) denotes the universal covering of \(M\).
This paper is concerned with the study of radiant affine structures on closed \(3\)-manifolds \(M\). The author classifies all such structures that verify one of the following conditions: (i) their holonomy group contains a solvable subgroup of finite index; (ii) their radial flow is tangent to a flat embedded surface of \(M\), (iii) their radial flow admits at least a periodic orbit which is not of hyperbolic type; (iv) the image of their developing mapping is an open convex set of \(\mathbb{R}^3\), that is a cone with the origin as vertex. For example, if \(M\) is a closed \(3\)-manifold endowed with an \((\text{GL}(3, \mathbb{R}), \mathbb{R}^3)\)-structure that satisfies condition (ii), then \( M \) is a generalized affine suspension [Y. Carrière, Questions ouvertes sur les variétés affines, Sémin. Gaston Darboux Géom. Topologie Différ. 1991-1992 , 69-72 (1993)]. Using the author’s results of this paper, first published as a preprint in 1997, S. Choi was able to obtain the complete classification of closed radiant affine \(3\)-manifolds [S. Choi, The decomposition and the classification of radiant affine \(3\)-manifolds (avec un appendice par Barbot-Choi), to appear in Mem. Am. Math. Soc.], answering a conjecture by Y. Carrière (loc. cit.).

MSC:

57M50 General geometric structures on low-dimensional manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53A20 Projective differential geometry

Citations:

Zbl 0767.53007
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References:

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