14 1. Various Ways of Representing Surfaces and Examples

The graph of f1 is the northern hemisphere, and the graph of f2

is the southern. However, we run into problems at the equator z = 0;

for reasons which will be made apparent when we give the precise

definition of a manifold (topological or differentiable), it is important

that the domain on which we define each graph be open. In this

particular case, this means we cannot include the equator in either

the northern or the southern hemisphere, and must cover those points

with other graphs. By using graphs with x or y as the dependent

variable, we can cover the ‘eastern’ and ‘western’ hemispheres, as

it were, but find that we require six graphs to deal with the entire

sphere, as shown in Figure 1.10.

This approach has wide validity. Recall that (x, y, z) ∈

R3

is

a critical point of a smooth function F :

R3

→ R if the gradient of

F vanishes at (x, y, z), and that a point is called regular if it is not

critical. If S is the zero set of such a function, then at any regular

point in S we can apply the Implicit Function Theorem and obtain

a neighbourhood of the point which is the graph of some function;

in essence, we are projecting patches of our surface to the various

coordinate planes in R3. If our surface contains only regular points,

this allows us to describe the entire surface in terms of these local

coordinates.

As indicated in the first lecture, if the gradient vanishes at a point,

the set of solutions may not look like a nice surface. A trivial example

is the sphere of radius zero,

x2

+

y2

+

z2

= 0; a more interesting

example is the cone

x2

+

y2

−

z2

= 0 near the origin.

b. Other ways of introducing local coordinates. From the geo-

metric point of view, the choice of planes involved in representing a

surface as the union of graphs of functions is somewhat arbitrary

and unnatural; for example, the orthogonal projection of the north-

ern hemisphere of S2 to the xy-plane represents points in the ‘arctic’

quite well, but distorts things rather badly near the equator, where

the derivative of the function blows up. If we are interested in an-

gles, distances, and other geometric qualities of the surface, a more

natural choice is to project to the tangent plane at each point; this

will lead us eventually to the notion of a Riemannian manifold. If

the previous approach represented an effort to draw a ‘world map’ of