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
10These
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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