14 1. Various Ways of Representing Surfaces and Examples The graph of f1 is the northern hemisphere, and the graph of f2 is the southern. However, we run into problems at the equator z = 0 for reasons which will be made apparent when we give the precise definition of a manifold (topological or differentiable), it is important that the domain on which we define each graph be open. In this particular case, this means we cannot include the equator in either the northern or the southern hemisphere, and must cover those points with other graphs. By using graphs with x or y as the dependent variable, we can cover the ‘eastern’ and ‘western’ hemispheres, as it were, but find that we require six graphs to deal with the entire sphere, as shown in Figure 1.10. This approach has wide validity. Recall that (x, y, z) R3 is a critical point of a smooth function F : R3 R if the gradient of F vanishes at (x, y, z), and that a point is called regular if it is not critical. If S is the zero set of such a function, then at any regular point in S we can apply the Implicit Function Theorem and obtain a neighbourhood of the point which is the graph of some function in essence, we are projecting patches of our surface to the various coordinate planes in R3. If our surface contains only regular points, this allows us to describe the entire surface in terms of these local coordinates. As indicated in the first lecture, if the gradient vanishes at a point, the set of solutions may not look like a nice surface. A trivial example is the sphere of radius zero, x2 + y2 + z2 = 0 a more interesting example is the cone x2 + y2 z2 = 0 near the origin. b. Other ways of introducing local coordinates. From the geo- metric point of view, the choice of planes involved in representing a surface as the union of graphs of functions is somewhat arbitrary and unnatural for example, the orthogonal projection of the north- ern hemisphere of S2 to the xy-plane represents points in the ‘arctic’ quite well, but distorts things rather badly near the equator, where the derivative of the function blows up. If we are interested in an- gles, distances, and other geometric qualities of the surface, a more natural choice is to project to the tangent plane at each point this will lead us eventually to the notion of a Riemannian manifold. If the previous approach represented an effort to draw a ‘world map’ of
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