20 1. Various Ways of Representing Surfaces and Examples

Figure 1.14. Determining distances in RP

2

via central angles.

be the same point, we obtain the quotient space

S2/I,

which is again

the projective plane, or the elliptic plane when we are interested in

the geometry.

In the elliptic plane, there is no such notion as the sign of an angle;

we cannot consistently determine which angles are positive and which

are negative. All the other geometric notions carry over, however; the

distance between two points can still be found as the magnitude of

the (acute) central angle they make (Figure 1.14), and the notions of

angle between geodesics and length of geodesics are still well defined.

Exercise 1.10. Write at least five propositions from Euclidean ge-

ometry which are true in the elliptic plane and at least three propo-

sitions which are true in Euclidean geometry and are not true in the

elliptic plane. Each proposition must include statements about con-

figurations of lines and/or isometries, and no two should be trivial

reformulations of each other.

b. A first glance at geodesics. Informally, as mentioned above,

a geodesic is a curve of shortest length between two points; more

precisely, it is a curve γ with the property that given any two points

γ(a) and γ(b) whose parameter values a and b are suﬃciently close

together, any other curve from one point to the other will have length

at least as great as the portion of γ between the two. Later in the

course (Lecture 25), we will consider the question of whether such a

curve always exists between two points, and whether it is unique.

The two most basic examples are the Euclidean spaces

Rn,

where

geodesics are straight lines, and the round sphere

S2,

where geodesics