Geodesics

In differential geometry, a geodesic is a curve that represents the shortest path between nearby points on a curved space, or more generally a locally distance-minimizing curve. Geodesics can also be described as critical points of the energy functional E(γ) = (1/2) ∫ |γ'(t)|^2 dt for curves γ with fixed endpoints. When parameterized by arc length, they are called unit-speed geodesics.

On a Riemannian manifold, geodesics satisfy the geodesic equation D/dt γ'(t) = 0, where D/dt is the covariant derivative along the curve and the connection is the Levi-Civita connection. In local coordinates this becomes a second-order ordinary differential equation involving Christoffel symbols Γ^k_{ij}.

Examples illustrate the concept. In Euclidean space, geodesics are straight lines. On the surface of a sphere, geodesics are great circles. On a cylinder, geodesics project to straight lines on the unrolled plane; when traced with arc length they appear as helices or straight lines depending on the direction.

Geodesics have the key property of locally minimizing length, though they need not be globally shortest between endpoints. Through any point with any initial velocity, there exists a geodesic defined for at least a short time, and geodesics depend smoothly on their initial data.

Applications include physics and geometry. In general relativity, geodesics describe the motion of free-falling particles in curved spacetime. More generally, they play a central role in the study of the intrinsic geometry of manifolds and in variational problems involving curves.