Geometric Manifolds

Abstract

In this chapter, we lay down the foundation for the theory of hyperbolic manifolds. We begin with the notion of a geometric space. Examples of geometric spaces are S ⁿ, E ⁿ, and H ⁿ. In Sections 8.2 and 8.3, we study manifolds locally modeled on a geometric space X via a group G of similarities of X. Such a manifold is called an (X, G)-manifold. In Section 8.4, we study the relationship between the fundamental group of an (X, G)-manifold and its (X, G)-structure. In Section 8.5, we study the role of metric completeness in the theory of (X, G)-manifolds. In particular, we prove that if M is a complete (X, G)-manifold, with X simply connected, then there is a discrete subgroup Γ of G of isometries acting freely on X such that M is isometric to X/Γ. The chapter ends with a discussion of the role of curvature in the theory of spherical, Euclidean, and hyperbolic manifolds.