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 Iz x = 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.