A COMPACTNESS THEOREM OF A SET OF ASPHERICAL RIEMANNIAN ORBIFOLDS

Abstract

This chapter presents a compactness theorem of a set of aspherical Riemannian orbifolds. The chapter also presents a theorem that states that for each positive number D and positive integer n, there exists a positive number v(n,D). In the case of aspherical orbifolds, the elements of the compactification are also an orbifold. A complete Riemannian manifold X can be geometrically contractible if, for each positive number r, there exists a positive number R(r) such that, for every point p of X, the inclusion map of Br (p,X) into BR(r) (p,X) is homotopic to the constant map.