48 1. Various Ways of Representing Surfaces and Examples

In the case R × R = R2, this corresponds to the fact that the com-

position of a flip about a vertical line with a flip about a horizontal

line is equivalent to rotation by π around the intersection of the two

lines.

With the geodesic flip defined, we can then ask whether the prod-

uct space X × Y is symmetric, and it turns out that if X and Y are

both symmetric spaces, so is their direct product X × Y . In this

manner we can obtain many higher-dimensional examples, and so if

we were to attempt to classify such spaces, we would want to focus

on those which are irreducible in that they cannot be decomposed as

the direct product of two lower-dimensional spaces, since the other

examples will be built from these.

The direct product provides a common means by which we de-

compose objects of interest into simpler examples in order to gain

a complete understanding. We find many examples of this in linear

algebra, in which context the phrase direct sum is also sometimes

used. Any finite-dimensional vector space can be written as the di-

rect product of n copies of R; this is just the statement that any

finite-dimensional vector space has a basis. A more sophisticated ap-

plication of this process is the decomposition of a linear transforma-

tion in terms of its action upon its eigenspaces, so that a symmetric

matrix can be written as the direct product of one-dimensional trans-

formations, while for a general matrix, we have the Jordan normal

form.

This process is also used in the classification of finitely generated

abelian groups, where we decompose the group of interest into a direct

sum of copies of Z and cyclic groups whose order is a power of a

prime, so that no further decomposition is possible. Thus the natural

counterpart to the study of how a particular sort of mathematical

structure can be decomposed is the study of what instances of that

structure are, in some appropriate sense, irreducible.