ABSTRACT
In Chapter 11 we found that every surface admits one of the three homogeneous two-dimensional geometries. The sphere and the projective plane admit spherical geometry, the torus and the Klein bottle admit flat geometry, and all other surfaces admit hyperbolic geometry. In Chapters 14, 15, and 16 we discussed the analogous three-dimensional geometries and found some examples of three-manifolds that admit them. It turns out that five more homogeneous geometries arise in the study of closed three-manifolds. One of them is the local geometry of S2 × E. S2xE geometry@S 2×E geometry (S2 × E, S2 × I, and S2 × S1 S2xS1@S 2×S 1 all have the same local geometry. It’s traditional to name the geometry after S2 × E because S2 × E is the “biggest” manifold having it.) This geometry is homogeneous homogeneous manifold manifold!homogeneous but not isotropic isotropic manifold manifold!isotropic . It’s homogeneous because it’s everywhere the same. But it’s not isotropic because at any given point we can distinguish some directions from others. Recall from Figure 6.10 that some cross-sections of S2 × S1 are spheres while others are flat tori. Locally one observes that some two-dimensional slices have positive curvature while others have zero curvature (Figure 18.1). The term sectional curvature sectional curvature refers to the curvature of a two-dimensional slice of a manifold. The word “section” comes from the latin “sectio” which means “slice” (more or less). Thus S2 × S1 has positive sectional curvature in the horizontal direction but zero sectional curvature in any vertical slice. An isotropic geometry has the same sectional curvature in all directions; the sectional curvatures of three-dimensional spherical geometry are all positive, those of three-dimensional Euclidean geometry are all zero, and those of three-dimensional hyperbolic geometry are all negative.
In Exercise 6.7 you found that P2 × S1 P2xS1@P 2×S 1 has S2 × E geometry. Name another nonorientable manifold with this geometry. (Hint: It first appeared in Chapter 17.)
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