46 1. Various Ways of Representing Surfaces and Examples distance along the geodesic from x as y is, but in the other direc- tion. It is immediate that this map preserves lengths along geodesics through x it may happen, however, that the distances between these geodesics vary, in which case the map would not be isometric. If the map is indeed isometric on some neighbourhood of x, and if this property holds for the geodesic flip Ix through any point x ∈ X, then we say that X is locally symmetric. The classification of such spaces (in any dimension) is one of the triumphs of Lie theory. No- tice that the geodesic flip may not be extendable to a globally defined isometry, so the isometry group of a locally symmetric space may be (and sometimes is) quite small. Although we have not yet encoun- tered any such examples, later on (Lecture 31) we will construct the hyperbolic octagon, whose isometry group can be shown to be finite, even though the space is locally symmetric. Given two nearby points x, y, we can take the point z lying at the midpoint of the geodesic segment connecting them. Then Izx = y. If X is connected (and hence path-connected) then any two points can be connected by a finite chain of neighbourhoods where these local isometries are defined. This implies that for any two points in a locally symmetric space, there exists an isometry between small enough neighbourhoods of those points. In other words, locally such a space looks the same near every point. If for any point x ∈ X the geodesic flip Ix can be defined not just locally, but globally (that is, extended to the entire surface X), and if it is in fact an isometry of X, then we say X is globally symmetric. In this case, the group of isometries Isom(X) acts transitively on all of X. In the previous lecture we discussed a related, but stronger, no- tion, in which we require Isom(X) to act transitively not only on points in X, but also on unit tangent vectors. If this holds, then in particular, given any x ∈ X, there is an isometry of X taking some tangent vector at x to its opposite this isometry must then be the geodesic flip, and so X is globally symmetric. It is not the case, however, that every globally symmetric space has this property of transitive action on tangent vectors the flat torus is one example.

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