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
Previous Page Next Page