40 1. Various Ways of Representing Surfaces and Examples

with determinant equal to positive one is the special orthogonal group

SO(3).

In order to see that the members of O(3) are in fact the isome-

tries of

S2,

we could take the synthetic approach and look at the

images of three points not all lying on the same geodesic, as we did

with

Isom(R2);

in particular, we can take the standard basis vectors

e1, e2, e3.

An alternate approach is to extend the isometry to

R3

by ho-

mogeneity. That is, given an isometry I :

S2

→

S2,

we can define a

linear map A:

R3

→

R3

by

Ax = x · I

x

x

.

It follows that A preserves lengths in R3, and in fact, this is suﬃcient

to show that it preserves angles as well. This can be seen using a

technique called polarisation, which allows us to express the inner

product in terms of the norm, and hence show the general result that

preservation of norm implies preservation of inner product:

x + y

2

= x + y, x + y

= x, x + 2 x, y + y, y

= x

2

+ y

2

+ 2 x, y ,

x, y =

1

2

( x + y

2

− x

2

− y

2).

This is a useful trick to remember, and it allows us to show that a

symmetric bilinear form is determined by its diagonal part. In our

particular case, it shows that the matrix A we obtained is in fact in

O(3), since it preserves both lengths and angles.

The matrix A ∈ O(3) has three eigenvalues, some of which may be

complex. Because A is orthogonal, we have |λ| = 1 for each eigenvalue

λ; further, because the determinant is the product of the eigenvalues,

we have λ1λ2λ3 = ±1. The entries of the matrix A are real, hence the

coeﬃcients of the characteristic polynomial are as well; this implies

that if λ is an eigenvalue, so is its complex conjugate

¯.

λ

There are two cases to consider. Suppose det(A) = 1. Then the

eigenvalues are λ,

¯,

λ and 1, where λ =

eiα

lies on the unit circle in