1997/06/23 by Suhyoung Choi, Choi, Suhyoung
Mathematics · #53C15(Secondary) #57M50 (Primary) 53A20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #dg-ga #math.DG #msc:53A20 #msc:57M50
paper · pdf · doi:10.48550/arxiv.dg-ga/9706011
67 pages, 27 figures, some revisions. To appear in the International Journal Of Mathematics
openalex publication_date 1997/06/23 · arxiv created 1999/12/21 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold M with holonomy group of shrinkable dimension (or discompacité in French) less than or equal to two is diffeomorphic to \bR3. Hence, M is irreducible. This follows from two results: (i) a simply connected affine 3-manifold which is 2-convex is diffeomorphic to \bR3, whose proof using the Morse theory takes most of this paper; and (ii) a closed affine manifold of holonomy of shrinkable dimension less or equal to d is d-convex. To prove (i), we show that 2-convexity is a geometric form of topological incompressibility of level sets. As a consequence, we show that the universal cover of a closed affine three-manifold with parallel volume form is diffeomorphic to \bR3, a part of the weak Markus conjecture. As applications, we show that the universal cover of a hyperbolic 3-manifold with cone-type singularity of arbitrarily assigned cone-angles along a link removed with the singular locus is diffeomorphic to \bR3. A fake cell has an affine structure as shown by Gromov. Such a cell must have a concave point at the boundary.