Lecture 7 45 (3) the flat Klein bottle, i.e. the square with appropriately iden- tified pairs of opposite sides. Consider a more general class of examples, which generalise the construction of the flat torus as R2/Z2. Let L be a lattice in R2—that is, a set of vectors of the form { mu + nv | m, n ∈ Z }, where u and v are two fixed linearly independent vectors. We can identify the factor space R2/L with the parallelogram { su + tv | 0 ≤ s, t ≤ 1 } with pairs of opposite sides identified by translations. Exercise 1.22. Show that the following statements hold. (1) The factor space R2/L is homeomorphic to a torus (2) R2/L has a natural metric which is locally isometric to R2 (3) The isometry group acts transitively on R2/L. The ‘crystallographic restriction’ property established in the fol- lowing exercise aids in the classification of isometries of these tori. Exercise 1.23. Show that any non-trivial isometry of R2/L with a fixed point has period 2, 3, 4, or 6. b. Symmetric spaces. The discussion of spaces with lots of isome- tries is related to the notion of a symmetric space, which we will now examine more closely. In what follows, we assume certain properties of geodesics which will be formally described (but not proved) later in this course. In particular, we assume that there is a unique geodesic passing through a given point in a given direction, and that there is a unique shortest geodesic connecting any two suﬃciently close points. Of course, all of this assumes the metric on our surface is given in a nice way, as has been the case with all examples considered so far.10 Given a point x on a surface X, we define the geodesic flip through x, denoted by Ix, as follows. For each geodesic γ passing through x, each point y lying on γ is sent to the point on γ which is the same 10 These notions of direction and ‘nice’ metrics, which are rather vague at the moment, will be made more precise when we discuss smooth manifolds and Riemannian metrics in Chapters 3 and 4.

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