Lecture 1 9

Figure 1.8. Meridians and parallels on two tori with different geometries.

obtain a surface resembling a torus as far as its global properties are

concerned. For example, vertical and horizontal segments become

closed curves which are identified with “parallels” and “meridians”

of the torus of revolution—this will become clear in the next lecture

when we introduce parametric representations of surfaces. However,

the geometry of our surface, the flat torus, is different from that of

the torus of revolution. For example, all vertical and all horizontal

“circles” in the flat torus have the same length, while in the torus

of revolution the meridians have the same length but the parallels

do not (Figure 1.8). This is a consequence of the fact that although

the cylinder in

R3

has the same intrinsic geometry as the sheet of

paper with only one pair of sides identified (that is, the paper is not

stretched), it cannot be bent in

R3

without a distortion. So far, our

notion of intrinsic geometry is intuitive, but soon we will make it more

precise.

Let us try to exhaust the possibilities of surface-building from a

rectangular piece of paper. The only remaining way of identifying

pairs of opposite sides is to identify both pairs of sides using a flip, so

that we identify (x, 0) with (1 − x, 1) and (0, y) with (1, 1 − y). We

will now turn our attention to this construction.

Exercise 1.3. Describe the surface obtained from the square by iden-

tifying points on pairs of adjacent sides, i.e. (0, t) with (1 − t, 1) and

(1, t) with (1−t, 0). Pay attention both to the shape and to geometry.

d. Projective plane and flat torus as factor spaces. To get a

more symmetric picture for the last construction, we may inflate the

square to a disc into which the square is inscribed, project the bound-

ary of the square radially to the circumference of the disc, and observe