38 1. Various Ways of Representing Surfaces and Examples

Here O(2) is the group of real-valued orthogonal 2 × 2 matrices, and

the plane upon which

Isom(R2)

acts is the horizontal plane z = 1 in

R3.

Exercise 1.16. Prove that every isometry of the Euclidean plane can

be represented as a product of at most three reflections.

Exercise 1.17. Consider all possible configurations of two and three

lines in the plane: two lines may be either parallel or intersecting;

for three lines there are a few more options. Identify the product of

reflections in those lines for each case as one of four types of isometries.

Exercise 1.18. Consider an orientation reversing isometry in the

complex form z → a¯ z + b. Find a condition on a, b ∈ C which will

determine if it is a reflection or a glide reflection, and identify the axis

in both cases.

b. Isometries of the sphere and the elliptic plane. By counting

dimensions in the isometry group of the Euclidean plane, we argued

that almost every orientation preserving isometry has a fixed point,

while almost every orientation reversing isometry has no fixed point.

In the next lecture, we will see that the picture for the sphere is

somewhat similar—now any orientation preserving isometry has a

fixed point, and most orientation reversing ones have none. For the

elliptic plane, however, it will turn out to be dramatically different:

any isometry has a fixed point, and can in fact be interpreted as a

rotation!

Many of the arguments in the previous section carry over to the

sphere; the same techniques of taking intersections of circles, etc.

still apply. The classification of isometries on the sphere is somewhat

simpler, since every orientation preserving isometry has a fixed point,

while every orientation reversing isometry (other than reflection in a

great circle) has a point of period two, which becomes a fixed point

when we pass to the elliptic plane.

We will be able to show that every orientation preserving isometry

of the sphere comes from a rotation of

R3,

and that the product of

two rotations is itself a rotation. This is slightly different from the

case with

Isom(R2),

where the product could either be a rotation, or

if the two angles of rotation summed to zero (or a multiple of 2π), a