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.

1For

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.