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.

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