An affine manifold is a manifold with torsion-free flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with a holonomy group consisting of affine transformations fixing a common fixed point. We decompose a closed radiant affine $3$-manifold into radiant $2$-convex affine manifolds and radiant concave affine $3$-manifolds along mutually disjoint totally geodesic tori or Klein bottles using the convex and concave decomposition of real projective $n$-manifolds developed earlier. Then we decompose a $2$-convex radiant affine manifold into convex radiant affine manifolds and concave-cone affine manifolds. To do this, we will obtain certain nice geometric objects in the Kuiper completion of a holonomy cover. The equivariance and local finiteness property of the collection of such objects will show that their union covers a compact submanifold of codimension zero, the complement of which is convex. Finally, using the results of Barbot, we will show that a closed radiant affine $3$-manifold admits a total cross-section, confirming a conjecture of Carrière, and hence every closed radiant affine $3$-manifold is homeomorphic to a Seifert fibered space with trivial Euler number, or a virtual bundle over a circle with fiber homeomorphic to a Euler characteristic zero surface. In Appendix C, Thierry Barbot and the author show the nonexistence of certain radiant affine $3$-manifolds and that compact radiant affine $3$-manifolds with nonempty totally geodesic boundary admit total cross-sections, which are key results for the main part of the paper.

Readership

Graduate students and research mathematicians interested in manifolds and cell complexes, and differential geometry.