Geodesic

Summary

A geodesic on a Riemannian manifold is a curve whose velocity is parallel along itself for the Levi-Civita connection. So it has zero acceleration within the manifold. Geodesics generalise straight lines: short pieces of a geodesic are the shortest curves between their endpoints. They are solutions of a second-order ordinary differential equation, so a geodesic is determined by its starting point and starting velocity. Geodesics are the basic tool for studying distance, curvature and topology of Riemannian manifolds.