Riemannian Geodesics

Abstract

If we are to use geodesies and covariant derivatives as tools for studying Riemannian geometry, it is evident that we need a way to single out a particular connection on a Riemannian manifold that reflects the properties of the metric. In this chapter, guided by the example of an embedded submanifold of R ⁿ, we describe two properties that determine a unique connection on any Riemannian manifold. The first property, compatibility with the metric, is easy to motivate and understand. The second, symmetry, is a bit more mysterious.