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
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