Geodesic

In geometry, a geodesic (/ˌəˈdɛsɪk, ˌ-, -ˈd-, -zɪk/[1][2]) is commonly a curve representing in some sense the shortest[lower-alpha 1] path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line" to a more general setting.

The noun geodesic[lower-alpha 2] and the adjective geodetic[lower-alpha 3] come from geodesy, the science of measuring the size and shape of Earth, while many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph.

In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishing geodesic curvature. More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion.

Geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles.