Lecture 1 3 Figure 1.2. Three ellipsoids. squeezing the sphere along the three coordinate axes produces an ellipsoid given by the equation (1.4) x2 a2 + y2 b2 + z2 c2 = 1, where a, b, and c are parameters which depend on the degree of stretching or squeezing. Of the three surfaces above, the overall shape and crude properties of an ellipsoid (its topology) are most similar to that of a sphere, and are quite different from that of a cylinder or a cone its geometry, however, displays many differences from the geometry of a sphere.1 For example, the sphere has many symmetries (that is, rigid motions of the space which leave the sphere as a whole in place), while a triaxial ellipsoid (one for which all three numbers a, b, and c in (1.4) are different, such as the third shape shown in Figure 1.2) has only a few. Exercise 1.2. Find all the symmetries for (1) a triaxial ellipsoid (2) an ellipsoid of revolution for which a = b = c (such as the second ellipsoid in Figure 1.2). Consider separately the symmetries which can be effected by a contin- uous motion of the space and those which cannot, such as reflections with respect to planes. 1 For the time being, we rely on intuitive ideas of what constitutes a general shape. For a reader steeped in mathematical rigor, we refer to notions of homeomorphism and diffeomorphism, which will be introduced later in Lectures 4 and 17, respectively, and say that two surfaces have similar shapes if they are homeomorphic, or diffeomorphic in the case of smooth surfaces.
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