Lens-shaped totally geodesic ends of convex real projective manifolds

Abstract:

Convex projective structures are generalizations of hyperbolic structures on n-manifolds (orbifolds). We will study totally geodesic ends of convex real projective n-manifolds. These are ends that we can compactify by totally geodesic orbifolds of codimension-one. We propose a sufficient condition for lens-shaped end-neighborhoods to exist for a totally geodesic end. The condition is that of the uniform-middle-eigenvalues on the end-holonomy-group. For this, we attempt to show that every affine deformation of a discrete dividing linear group satisfying this condition acts on properly convex domains in the affine n-space. We will mention about the lens-shaped radial ends which are dual to lens-shaped totally geodesic ends. We also discuss the relationship to the globally hyperbolic space-times in flat Lorentz geometry.