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
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
by approximating
it with piecewise linear paths and then using the notion of distance
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