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.

Lecture 3

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

R3,

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

R3

without self-intersections

described above.