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