Chapter 3 Geodesics

Abstract

A geodesic on a Riemannian manifold is the analogue of a line in Euclidean space. One can characterize a line in Euclidean space in several equivalent ways, among which are the following:

• (1) A line is “straight” in the sense that it has a parametrization with a constant velocity vector field.

• (2) A line connecting two points gives the shortest distance between the two points.

These two properties are not necessarily equivalent on a Riemannian manifold. We define a geodesic by generalizing the notion of “straightness.” For this, it is not necessary to have a metric, but only a connection. On a Riemannian manifold, of course, there is always the unique Riemannian connection, and so one can speak of geodesics on a Riemannian manifold.