26 1. Various Ways of Representing Surfaces and Examples

a vertex; at a vertex, the angles add up to less than 2π, whereas

everywhere else, they add up to exactly 2π. However, we cannot

tell whether or not we are at an edge; this has to do with the fact

that given two points on adjacent faces, the way to find the shortest

path between them is to unfold the two faces and place them flat

on the plane (at which stage points on an edge look just like points

on a face), draw a straight line between the two points in question,

and then fold the surface back up (Figure 1.18). As far as our two-

dimensional selves are concerned, points on an edge and points on a

face are indistinguishable, since the unfolding process does not change

any distances along the surface.

It is also possible that a surface which can be embedded in

R3

will

lose some of its nicer properties in the process. For example, the usual

embedding of the torus destroys the symmetry between meridians and

parallels; all of the meridians are the same length, but the length of

the parallels varies. We can retain this symmetry by embedding in

R4, the so-called flat torus. Parametrically, this is given by

x = r cos t, y = r sin t,

z = r cos s, w = r sin s,

where s, t ∈ [0, 2π]. As we already mentioned, we can also obtain

the flat torus as a factor space, using the same method as in the

definition of the projective plane or Klein bottle. Beginning with a

rectangle, we identify opposite sides (with no reversal of direction);

alternately, we can consider the family of isometries of R2 given by

Tm,n : (x, y) → (x + m, y + n), where m, n ∈ Z, and mod out by

orbits. This construction of T2 as R2/Z2 is exactly analogous to the

construction of the circle S1 as R/Z.

We have seen that surfaces can be considered from different view-

points: sometimes we treat them as geometric objects, with intrinsi-

cally defined distances, angles, and areas, while other times we treat

them as ‘stretchable’ objects which can be bent and deformed, but

not torn or broken. In mathematical language, this corresponds to

considering different structures on surfaces, and this is the central

theme of this course, which we will take up in earnest in the next

lecture.