A classification of radial and totally geodesic ends of properly convex real projective orbifolds III: the convex but nonproperly convex and non-complete-affine radial ends

Abstract

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The purpose of this paper is to understand the structures of ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. In previous papers, we classified properly convex or complete radial ends under suitable conditions. In this paper, we will study radial ends that are convex but not properly convex nor complete affine. The main techniques are the theory of Fried and Goldman on affine manifolds, and a generalization of the work on Riemannian foliations by Molino, Carri\`ere, and so on. We will show that these are quasi-joins of horospheres and totally geodesic radial ends. These are deformations of joins of horospheres and totally geodesic radial ends.