Lecture 7 43

opposite them, which have been painted all three colours. (We neglect

the boundaries of the wedges, since they have area zero.) Hence if we

add up the areas of the double wedges, we obtain

areas of wedges = blue area + yellow area + red area

= (area of sphere) + 4 × (area of triangle),

which allows us to write an equation for the area A of the triangle:

4(α + β +

γ)R2

=

4πR2

+ 4A.

Solving, we see that

(1.6) A =

R2(α

+ β + γ − π).

Thus the area of the triangle is directly proportional to its angular

excess; this result has no analogue in planar geometry, due to the

flatness of the Euclidean plane. As we will see later on in the course,

it does have an analogue in the hyperbolic plane, where the angles of

a triangle add up to less than π, and the area is proportional to the

angular defect.

Exercise 1.20. Express the area of a geodesic polygon on the sphere

in terms of its angles.

Lecture 7

a. Spaces with lots of isometries. In our discussion of the isome-

tries of R2, S2, and RP 2, we have observed a number of differences

between the various spaces, as well as a number of similarities. One of

the most important similarities is the high degree of symmetry each

of these spaces possesses, as evidenced by the size of their isometry

groups.

We can make this a little more concrete by observing that the

isometry group acts transitively on each of these spaces; given any

two points a and b in the plane, on the sphere, or in the projective

plane, there is an isometry I of the space such that Ia = b.

In fact, we can make the stronger observation that the group acts

transitively on the set of unit tangent vectors. That is to say, if v is

a unit tangent vector at a, which can be thought of as indicating a

particular direction along the surface from the point a, and w is a