Lecture 6 41
the complex plane. Let x be the eigenvector corresponding to the
eigenvalue 1, and note that A acts on the plane orthogonal to x by
rotation by α; hence A is a rotation by α around the axis through x.
The second case, det(A) = −1, can be dealt with by noting that A
can be written as a composition of −I (reflection through the origin)
with a matrix with positive determinant, which must be a rotation,
by the above discussion. Upon passing to the elliptic plane RP
reflection −I becomes the identity, so that every isometry of RP
a rotation.
This result, that every isometry of the sphere is either a rotation
or the composition of a rotation and a reflection through the origin,
shows that every isometry has either a fixed point or a point of period
two, which becomes a fixed point upon passing to the quotient space
RP 2.
As a concrete example of how all isometries become rotations in
RP 2, consider the map A given by reflection through the xy-plane,
A(x, y, z) = (x, y, −z). Let R be rotation by π about the z-axis, given
by R(x, y, z) = (−x, −y, z). Then A = R (−I), so that as maps on
RP 2, A and R coincide. Further, any point (x, y, 0) on the equator
of the sphere is fixed by this map, so that R fixes not only one point
in RP 2, but many.
Exercise 1.19. Let x and y be two points in the elliptic plane.
(1) Prove that there are at most two shortest curves connecting
x and y.
(2) Find a necessary and sufficient condition for uniqueness of
the shortest curve connecting x and y.
b. Area of a spherical triangle. In the Euclidean plane, the most
symmetric formula for determining the area of a triangle is Heron’s
A = s(s a)(s b)(s c),
where a, b, c are the lengths of the sides, and s =
(a + b + c) is
the semiperimeter of the triangle. There are other, less symmetric,
formulae available to us if we know the lengths of two sides and the
measure of the angle between them, or two angles and a side; if all
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