Parallel propagation around finite curves

Figure 5.1 repeats the previous example of non-Euclidean geometry on the surface of a sphere which we discussed in Section 2.2 of Chapter 2. We have the triangle ABC of Figure 2.3 whose three angles are each 90°. Consider what happens to a vector (shown by a dotted arrow) as it is parallely transported along the three sides of this triangle. As shown in Figure 5.1, this vector is originally perpendicular to AB when it starts its journey at A. When it reaches B it lies along CB; it keeps pointing along this line as it moves from B to C. At C it is again perpendicular to AC. So, as it moves along CA from C to A, it maintains this perpendicularity, with the result that when it arrives at A it is pointing along AB. In other words, one circuit around this triangle has resulted in a change of direction of the vector by 90°, although at each stage it was being moved parallel to itself!

A similar experiment with a triangle drawn on a flat piece of paper will tell us that there is no resulting change in the direction of the vector when it moves parallel to itself around the triangle. So our spherical triangle behaves differently from the flat Euclidean triangle.

The phenomenon illustrated in Figure 5.1 can also be described as follows.