Lecture 6 39

translation. We will, in fact, be able to obtain Isom(S2) as a group of

3 × 3 matrices in a much more natural way than we did for

Isom(R2)

above, since any isometry of

S2

extends to a linear orthogonal map

of

R3,

and so we will be able to use linear algebra directly.

Lecture 6

a. Classification of isometries of the sphere and the elliptic

plane. There are two approaches we can take to investigating isome-

tries of the sphere

S2;

we saw this dichotomy begin to appear when

we examined

Isom(R2).

The first is the synthetic approach, which

treats the problem using the tools of solid geometry; this is the ap-

proach used by the Greek geometers of late antiquity in developing

spherical geometry for use in astronomy.

The second approach, which we will follow below, uses methods of

linear algebra; translating the question about geometry to a question

about matrices puts a wide range of techniques at our disposal, which

will prove enlightening, and rather more useful now than it was in the

case of the plane, when the relevant matrices were only 2 × 2.

The first important result is that there is a natural bijection

(which is in fact a group isomorphism) between Isom(S2) and O(3),

the group of real orthogonal 3 × 3 matrices. The latter is defined by

O(3) = { A ∈ M3(R) |

AT

A = I }.

That is, O(3) comprises those matrices for which the transpose and

the inverse coincide. This has a nice geometric interpretation; we

can think of the columns of a 3 × 3 matrix as vectors in R3, so that

A = (a1|a2|a3), where ai ∈ R3. (In fact, ai is the image of the ith basis

vector ei under the action of A.) Then A lies in O(3) iff {a1, a2, a3}

forms an orthonormal basis for R3, that is, if ai, aj = δij , where

·,· denotes the inner product, and δij is the Kronecker delta, which

takes the value 1 if i = j, and 0 otherwise. The same criterion applies

if we consider the rows of A rather than the columns.

Since

det(AT

) = det(A), any matrix A ∈ O(3) has determi-

nant ±1; the sign of the determinant indicates whether the map pre-

serves or reverses orientation. The group of real orthogonal matrices