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4 1. THE RICCI FLOW OF SPECIAL GEOMETRIES AT3 is closed, irreducible, and homotopic to a hyperbolic 3-manifold, then J\f3 is homeomorphic to a hyperbolic 3-manifold. 2. Model geometries A geometric structure on an n-dimensional manifold Mn may be regarded as a complete locally homogeneous Riemannian metric g. A metric is locally homogeneous if it looks the same at every point. More precisely, one says that g is locally homogeneous if for all x, y G Mn there exist neighborhoods Ux C Mn of x and Uy C Mn of y and a g-isometry j x y : Ux — Uy such that j x y (x) = y. In this section, we shall develop a more abstract way to think about geometric structures. One says that a Riemannian metric g on Mn is homogeneous (to wit, globally homogeneous) if for every x,i/G A4n there exists a p-isometry Jxy • M71 ~* M.n such that j x y {x) = y. These concepts are equivalent when M.n is simply connected. PROPOSITION 1.7 (Singer [120]). Any locally homogeneous metric g on a simply-connected manifold is globally homogeneous. We also have the following well-known fact. PROPOSITION 1.8. If (Mn,g) is homogeneous, then it is complete. Thus if {Mn, g) is locally homogeneous, we say that its geometry is given by the homogeneous model (.Mn, #), where Mn is the universal cover of Mn and g is the complete metric obtained by lifting the metric g. Thus to study geometric structures, it suffices to study homogeneous models. A model geometry in the sense of Klein is a tuple (A1n, Q, ?*), where Ain is a simply-connected smooth manifold and Q is a group of diffeo- morphisms that acts smoothly and transitively on M71 such that for each x G A4n, the isotropy group (point stabilizer) is isomorphic to Q*. We say a model geometry (A4n,G,G*) is a maximal model geometry if Q is maximal among all subgroups of the diffeomor- phism group 2) (Mn) that have compact isotropy groups. (An example of particular relevance occurs when Mn = Q is a 3-dimensional unimodular Lie group. These maximal models were classified by Milnor in [98] and are briefly reviewed in Section 4 of this chapter.) The concepts of model geometry and homogeneous model are essentially equivalent (up to a subtle ambiguity that will be addressed in Remark 1.13). One direction is easy to establish. LEMMA 1.9. Every model geometry (Mn,G,G*) may be regarded as a complete homogeneous space (Mn y g). PROOF. One may obtain a complete homogeneous ^-invariant Riemann- ian metric on Mn as follows. Choose any x G Mn. ligx is any scalar product