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 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
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