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

2,

the

reflection −I becomes the identity, so that every isometry of RP

2

is

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

formula

A = s(s − a)(s − b)(s − c),

where a, b, c are the lengths of the sides, and s =

1

2

(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