32 1. Various Ways of Representing Surfaces and Examples

Figure 1.19. A planar model on a hexagon.

In fact, Isom(X, d) is not just a set; it has a natural binary oper-

ation given by composition, under which is becomes a group. This is

an example of a very natural and general kind of group which is often

of interest; all the bijections are from some fixed set to itself, with

composition as the group operation. On a finite set, this gives the

symmetric group Sn, the group of permutations. On an infinite set,

the group of all bijections becomes somewhat unwieldy, and it is more

natural to consider the subgroup of bijections which preserve a partic-

ular structure, in this case the metric structure of the space. Another

common example of this is the general linear group GL(n, R), which

is the group of all bijections from

Rn

to itself preserving the linear

structure of the space.

In the next lecture, we will discuss the isometry groups of Eu-

clidean space and of the sphere.

Exercise 1.15. Consider a regular hexagon with pairs of opposite

sides identified by the corresponding translations, as in Figure 1.19.

(1) Prove that it is a torus.

(2) Prove that locally, it is isometric to Euclidean plane.

(3) Prove that it is not isometric to the standard flat torus.

c. Other notions of dimension. As mentioned above, we usually

think of dimension as a topological invariant. However, for general

compact metric spaces there is another notion of dimension which is

a metric invariant, rather than a topological one. The main idea is

to capture the rate at which volume (or some other kind of measure)

scales with the metric; for example, a cube in

Rn

with side length r

has volume

rn,

and the exponent n is the dimension of the space.