10 1. Various Ways of Representing Surfaces and Examples
Figure 1.9. Various models for the real projective plane.
that the identified pairs become antipodal points on the boundary cir-
cle. Thus our object becomes the disc with pairs of opposite points
on the boundary identified, as in Figure 1.9. To make this even more
symmetric, inflate the disc to a hemisphere, keeping the boundary as
the equator. Now we can add the other hemisphere and observe that
each point of our object is represented by a pair of opposite points
on the sphere.
Instead of taking pairs of antipodal points as the points of our
surface, we may observe that any such pair determines a unique line in
R3
passing through the centre of the sphere, and vice versa. Thus we
may also think of our surface as the set of all lines through a particular
point—the surface so obtained is called the projective plane, denoted
RP
2.
An obvious advantage of the sphere representation over gluing
is that it highlights the uniformity of the surface; all points look the
same.
Inspired by the last construction, we may try to look at the flat
torus differently. First recall that the circle can be represented either
by an interval, say [0, 1], with endpoints identified, or as the set of
equivalence classes of real numbers modulo one, i.e. the set of all
fractional parts of real numbers. If we simply think of all numbers
with the same fractional part as the same element of the circle we
come to the representation S1 = R/Z—note that here every point on
the circle is represented in the same way, in contrast to the interval
with endpoints identified, where the choice of representation led to a
false distinction between endpoints and non-endpoints. This choice
of representation is a matter of fixing a fundamental domain; that is,
a subset of R which contains exactly one element of each equivalence
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