The Rigidity Theorem (Compact Case)

Abstract

In this chapter we are going to prove that there is a very sharp difference between 2-dimensional hyperbolic geometry and higher dimensions (at least for the compact case, but the results we shall prove generalize to the case of finite volume). Namely, we shall prove that for n ≥ 3 a connected, compact, oriented n-manifold supports at most one (equivalence class of) hyperbolic structure (while it was proved in Chapt. B that a compact surface of genus at least 2 supports uncountably many non-equivalent hyperbolic structures). This is the famous Mostow rigidity theorem: the original proof can be found in [Mos], and others (generalizing the first one) in [Mar] and [Pr]; we shall refer mostly to [Gro3], [Th1, ch. 6] and [Mu]. The core of the proof we present resides in Theorem C.4.2, relating the Gromov norm (introduced in C.3) to the volume of a compact hyperbolic manifold; this result has a deep importance independently of the rigidity theorem: in Chapters E and F we shall meet interesting applications and related ideas.KeywordsHyperbolic ManifoldKlein BottleRigidity TheoremGroup IsomorphismHyperbolic StructureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.