The decomposition and classification of radiant affine 3-manifolds (Q2753256)

An affine manifold is a manifold with an affine structure; i.e. an atlas of charts to the affine space \(\mathbb{R}^n\) with affine transition functions. An affine manifold is a radiant affine manifold if an atlas can be chosen so that the transition functions belong to the subgroup \(\text{GL}(n, \mathbb{R})\) of \(\text{Aff} (\mathbb{R}^n)\). Examples of radiant affine \((n+1)\)-manifolds can be obtained by generalized affine suspensions over real projective \(n\)-manifolds. In this context, there is a natural definition of totally geodesic submanifolds and convex subsets.NEWLINENEWLINENEWLINEIn this memoir a complete classification of compact radiant affine 3-manifolds with empty or totally geodesic boundary is obtained. The main result states that every compact radiant affine 3-manifold with empty or totally geodesic boundary decomposes (non-canonically) along the union of finitely many disjoint totally geodesic tori or Klein bottles, tangent to the radial flow, into either (i) convex radiant affine 3-manifolds or (ii) generalized affine suspensions of real projective spheres, real projective planes, real projective hemispheres, or \(\pi\)-annuli (or Möbius bands) of type \(C\) (see definition in Section 3 of [\textit{S. Choi}, J. Differ. Geom. 40, No. 2, 239-283 (1994; Zbl 0822.53009)]); or affine tori, affine Klein bottles, or affine annuli (or Möbius bands) with geodesic boundary. As a consequence of this decomposition theorem and of some results of T. Barbot, a conjecture formulated by \textit{Y. Carrière} in [Questions ouvertes sur les variétés affines, Sémin. Gaston Darboux Géom. Topologie Différ. 1991-1992, 69-72 (1993)] who asked whether every compact radiant affine 3-manifold admits a total cross section to the radial flow, can be answered in the affirmative. This implies that every compact radiant affine 3-manifold with empty or totally geodesic boundary is homeomorphic to a Seifert space with Euler number zero or a virtual bundle over a circle with fiber homeomorphic to a Euler characteristic zero compact surface.