The convex real projective manifolds and orbifolds with radial ends: the openness of deformations

A real projective orbifold is an n-dimensional orbifold modeled on the real projective space with groups PGL(n+1,R). We concentrate on orbifolds with a compact codimension 0 submanifold whose complement is a union of neighborhoods of ends diffeomorphic to (n-1)-dimensional orbifolds times intervals. A real projective orbifold has radial ends if each of its ends is foliated by projective geodesics concurrent to one another. It is said to be convex if any path can be homotoped to a projective geodesic with endpoints fixed. A real projective structure sometimes admits deformations to inequivalent parameters of real projective structures. We will prove the local homeomorphism between the deformation space of real projective structures on such an orbifold with radial ends with various conditions with the PGL(n+1,R)-representation space of the fundamental group with corresponding conditions. We will use a Hessian argument to show that a small deformation of a real projective orbifold with ends will remain properly and strictly convex in a generalized sense if so is the beginning real projective orbifold, provided that the ends behave in a convex manner. Here, we have to restrict each end to have a fundamental group isomorphic to a finite extension of a product of hyperbolic groups and abelian groups. The understanding of the ends is not accomplished in this paper as this forms an another subject. We will also prove the closedness of the convex real projective structures on orbifolds with irreducibilty conditions.