Geodesic

A geodesic is a curve that represents the shortest path between nearby points on a surface or, more generally, a curve that follows the straightest possible direction in a space equipped with a metric. In differential geometry, a geodesic can be defined as a curve whose tangent vector is parallel transported along itself, or equivalently as a solution to the geodesic equation, which involves the connection determined by the space’s metric. Intuitively, geodesics generalize the notion of straight lines to curved spaces.

On a surface embedded in Euclidean space, geodesics have zero geodesic curvature. They locally minimize length, but a geodesic segment need not be the globally shortest path between its endpoints. In the plane, geodesics are straight lines; on a sphere, geodesics are great circles such as the equator or meridians.

Geodesics also play a central role in physics. In general relativity, free-fall motion of particles follows timelike geodesics of spacetime, while light follows null geodesics. The paths are determined by the spacetime metric and the Levi-Civita connection through the geodesic equation.

Key properties include local length minimization and the possibility that a geodesic cannot be extended for all time, depending on the geometry of the space. With suitable conditions such as completeness, geodesics can be extended to maximal domains. Given a point and an initial tangent direction, there is a unique geodesic defined for a short interval, providing a foundation for much of Riemannian and Lorentzian geometry.