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
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
without a distortion. So far, our
notion of intrinsic geometry is intuitive, but soon we will make it more
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
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