Closed flat affine 3-manifolds are prime

Abstract

An (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space ${\mathbf R}^3$ with transition maps in the affine transformation group $Aff({\mathbf R}^3)$. Equivalently an affine $3$-manifold is a $3$-manifold with a flat torsion-free affine connection. We show that a closed affine $3$-manifold is either irreducible or is finitely covered by an affine Hopf manifold. A real projective $3$-manifold is a manifold with an atlas of charts to a real projective space ${\mathbf R} P^3$ with transition maps in the projective transformation group $PGL(4, {\mathbf R})$. Using the convex concave decomposition of real projective manifolds, we will show that a closed real projective $3$-manifold decomposes into concave affine submanifolds, toral $\pi$-submanifolds and $2$-convex real projective manifolds.