In this chapter we introduce the parallel transport of a vector along a curve and geodesics. We start by describing these notions for embedded surfaces and then extend them to abstract manifolds.

In the Euclidean plane, the shortest path between a pair of points is the straight line joining the two points. Let us now consider an embedded surface S in R3. Given two points on the surface, we would like to find a path between the points that minimizes the arc length. Such a path is not, in general, the straight line between the points, since this line may not lie on the surface. If the surface is compact, one can show that there exists a curve segment lying on the surface, which has the minimum arc length when compared to all other curve segments between the two points. Such a curve segment is called a minimizing geodesic. If the surface is not compact, minimizing geodesics are defined at least locally: any point on the surface has a neighborhood such that there exists an arc length minimizing curve segment from that point to every other point in that neighborhood. A geodesic is a smooth curve whose sufficiently short segments are all minimizing geodesics. For example, the geodesics of a sphere are the great circles.