18 1. Various Ways of Representing Surfaces and Examples
in R3, which we already know. If our surface is not embedded in
Euclidean space, however, we must replace this with an infinitesimal
notion of distance, the Riemannian metric alluded to above. We will
give a precise definition and discuss examples and properties of such
metrics later in this course.
a. More about the M¨ obius strip and projective plane. Let us
go back to the M¨ obius strip. The most common way of introducing
it is as a sheet of paper (or belt, carpet, etc.) whose ends have been
attached after giving one of them a half-twist. In order to represent
this surface parametrically, it is useful to consider the factor space
construction, which was discussed in the first lecture for the Klein
bottle and the flat torus, and which is even simpler in the case of the
M¨ obius strip.
Begin with a rectangle R. We are going to identify each point on
the left-hand vertical boundary of R with a point on the right-hand
boundary; if we identify each point with the point directly opposite to
it (on the same horizontal line), we obtain a cylinder. To obtain the
M¨ obius strip, we identify the lower left corner with the upper right
corner and then move inwards; in this fashion, if R = [0, 1] × [0, 1],
the point (0, t) is identified with the point (1, 1 − t) for 0 ≤ t ≤ 1.
To embed this in
we can effect the half-twist by a continuous
uniform rotation of an interval (the vertical lines in the model) whose
centre moves around a closed curve (say a circle), and which remains
perpendicular to that circle. Using the x-coordinate in the model as
the angular coordinate along the circle, and the y-coordinate as the
distance along the interval, one can write a parametric representation
of a M¨ obius strip in R3 (see Figure 1.5).
Exercise 1.9. Write explicit expressions for the parametric represen-
tation of a M¨ obius strip embedded into