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
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
however, will not generally lead
to isometries of
since they will usually fail to preserve the lattice
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.
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;