Lecture 2 11
class, except along its boundary, where it may contain two or more. In
this case, we may take any unit interval as our fundamental domain.
A similar observation may be made with two variables, where we
observe that the (flat) torus
T2
can be identified with the set of pairs
of fractional parts of real numbers:
T2
=
R2/Z2,
where
Z2
is the lattice of vectors with integer coordinates. These
equivalence classes are represented by points in the unit square (the
fundamental domain), once pairs of boundary points whose difference
is an integer have been identified.
We may make one further step into abstraction; instead of vectors
with integer coordinates, think about translations by those vectors.
Then each equivalence class in
R2/Z2
becomes an orbit of the group
of such translations acting on
R2,
and our factor space (or quotient
space) naturally becomes the space of orbits.
The same approach may be taken with the projective plane—
notice that the flip on the sphere is a transformation which generates
a group of two elements, since its square is the identity. The orbit
of a point under the action of this group consists of the point itself,
together with its antipode—identifying each such pair of points yields
the projective plane, which can thus be thought of as the space of
orbits of this twoelement group acting on the sphere.
Exercise 1.4. Represent the cylinder, the infinite M¨ obius strip, and
the Klein bottle as orbit spaces for some groups acting on the Eu
clidean plane
R2.
The infinite M¨ obius strip is the infinite rectangle
[0, 1] × R with each pair of points (0, y) and (1, −y) identified.
Lecture 2
a. Equations for surfaces and local coordinates. Consider the
problem of writing an equation for the torus; that is, finding a function
F :
R3
→ R such that the torus is the solution set {(x, y, z) ∈
R3

F (x, y, z) = 0}. Because the torus is a surface of revolution, we begin
with the equation for a circle in the xzplane with radius 1 and centre
at (2, 0):
S1
= (x, z) ∈
R2
(x −
2)2
+
z2
= 1 .