Geometric Manifolds

  • John G. Ratcliffe
Part of the Graduate Texts in Mathematics book series (GTM, volume 149)


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.


Fundamental Group Cauchy Sequence Hyperbolic Manifold Lens Space Geodesic Line 
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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • John G. Ratcliffe
    • 1
  1. 1.Department of MathematicsVanderbilt UniversityNashvilleUSA

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