44 1. Various Ways of Representing Surfaces and Examples

unit tangent vector at b, then not only can we find an isometry that

carries a to b, but we can find one that carries v to w.

Another example of a surface with this property is the hyper-

bolic plane, which will appear in Chapter 4, and has the remarkable

property that its isometry group allows not one but three natural

representations as a matrix group (or a factor of such a group by its

two-element centre).

In fact, these four examples are the only surfaces for which isome-

tries act transitively on unit tangent vectors. There are of course a

number of higher-dimensional spaces with this property: Euclidean

spaces, spheres, and projective spaces, which are all analogues of their

two-dimensional counterparts, immediately come to mind, and there

are many more besides.

As an example of a space for which this property fails, consider

the flat torus T2 = R2/Z2. The property holds locally, in the neigh-

bourhood of a point, but does not hold on the entire space. While

Isom(T2) acts transitively on points, it does not act transitively on

tangent vectors; some directions lie along geodesics which are closed

curves, while other directions do not. Another example is given by

the cylinder, and examples of a different nature will appear later when

we consider the hyperbolic plane and its factors.

What sorts of isometries does T2 have? We may consider trans-

lations z → z + z0; rotations of

R2,

however, will not generally lead

to isometries of

T2,

since they will usually fail to preserve the lattice

Z2.

The rotation by π/2 about the origin is permissible, as are the

flips around the x- and y-axes, and around the line x = y.

In general,

Z2

must be mapped to itself or a translation of itself,

and so the isometry group is generated by the group of translations,

along with the symmetry group of the lattice. The latter group is

simply D4, the dihedral group on four letters, which arises as the

symmetry group of the square.

Exercise 1.21. Describe all the isometries of

(1) the ‘hexagonal’ torus of Exercise 1.15;

(2) the flat M¨ obius strip;