28 1. Various Ways of Representing Surfaces and Examples

that it have maximal rank; the matrix is given by

Df =

⎛

⎝∂sf2

∂sf1 ∂tf1

∂tf2⎠

∂sf3 ∂tf3

⎞

and so our requirement is that the two tangent vectors to the surface,

given by the columns of Df, be linearly independent. Under this con-

dition, the Implicit Function Theorem guarantees that the parametric

representation is locally bijective and that its inverse is differentiable.

Parametric representations may of course have singularities. A

good example is the representation of the sphere given by the inverse

map to the geographic coordinates, which maps an open disc regularly

onto the sphere with a point removed, and collapses the boundary of

the disc into this single point.

Lecture 4

a. Remarks on metric spaces and topology. Geometry in its

most immediate form deals with measuring

distances.8

For this rea-

son, metric spaces are fundamental objects in the study of geometry.

In the geometric context, the distance function itself is the object of

interest; this stands in contrast to the situation in analysis, where

metric spaces are still fundamental (as spaces of functions, for exam-

ple), but where the metric is introduced primarily in order to have a

notion of convergence, and so the topology induced by the metric is

the primary object of interest, while the metric itself stands somewhat

in the background.

A metric space is a set X, together with a metric, or distance

function, d: X × X → R0

+,

which satisfies the following axioms for all

values of the arguments:

(1) Positivity: d(x, y) ≥ 0, with equality iff x = y.

(2) Symmetry: d(x, y) = d(y, x).

(3) Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z).

8The

reader should be aware, however, that in modern mathematical terminology,

the word ‘geometry’ may appear with adjectives like ‘aﬃne’ or ‘projective’. Those

branches of geometry study structures which do not involve distances directly.