Lecture 2 17 system, but rather uses several patches to accomplish the task. A parametric representation, on the other hand, usually involves a map from a plane domain to a surface which is onto, or at at least nearly so, as in the inverse to the stereographic projection. One should also keep in mind that, while the notion of an atlas of local coordinate systems has a precise meaning which we will describe in Chapter 3, the notion of parametric representation is somewhat vague. Exercise 1.8. Write a parametric representation of the torus of rev- olution (1.5) using the ‘latitude’ (position of a plane section) and ‘longitude’ (the angular coordinate along a plane section) as parame- ters. Use this representation to construct a bijection between the flat torus from Lecture 1(d) and the torus of revolution. d. Metrics on surfaces. As our discussion of local coordinates sug- gested, we must address the question of how the distance between two points on a surface is to be measured. In the case of the Euclidean plane, we have a formula, obtained directly from the Pythagorean theorem. For points on the sphere of radius R we also have a for- mula: the distance between two points is simply the angle they make with the centre of the sphere, multiplied by R. Properties of this dis- tance, such as the triangle inequality, can be deduced via elementary geometry, or by representing the points as vectors in R3 and using properties of the inner product. These explicit formulae are serendipitous consequences of the ex- tremely symmetric shapes of the plane and the sphere. What is the correct notion of distance on an arbitrary surface? Recalling that in the plane at least, the shortest path between two points is a straight line, and it is precisely along this line that the distance given by the Pythagorean theorem is measured, we may suggest that the distance between two points should naturally be defined as the length of the shortest path connecting them. In general, since we do not yet know whether such a shortest path always exists, the proper definition of distance is as the infimum of the set of lengths of paths connecting the two points. Of course, this requires that we have a definition for the length of a path on the surface. We can find the length of a path in R3 by approximating it with piecewise linear paths and then using the notion of distance
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