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Geodesics are curves that parallel-transport their own velocity — formally, ∇_{γ'} γ' = 0. They generalize straight lines to curved spaces: on a Riemannian manifold, geodesics are locally length-minimizing, and they are the critical points of the energy functional. The geodesic equation is a system of second-order ODEs whose solutions are determined by an initial point and initial velocity. Great circles on spheres, straight lines in Euclidean space, and freefall trajectories in general relativity are all geodesics.
In Euclidean space, straight lines are characterized in three equivalent ways: (1) they minimize distance, (2) they have zero acceleration, and (3) they parallel-transport their own velocity. On a Riemannian manifold, these three properties continue to characterize the same class of curves — geodesics — but the equivalence is nontrivial and depends on the connection.
The geodesic equation ∇_{γ'} γ' = 0 says that the covariant acceleration of γ vanishes. In coordinates: d²γᵏ/dt² + Γᵏᵢⱼ (dγⁱ/dt)(dγʲ/dt) = 0. The Christoffel symbol term is the "centripetal" correction that accounts for the curving of coordinate lines — on a sphere, great circles satisfy this equation even though their coordinate expressions are curved in (θ, φ) coordinates. Solutions exist and are unique for given initial point p = γ(0) and initial velocity v = γ'(0) ∈ TpM, by the existence and uniqueness theorem for ODEs.
Geodesics are also the critical points of the energy functional E(γ) = ½∫₀¹ |γ'(t)|² dt. The Euler-Lagrange equation for this variational problem is exactly the geodesic equation. Using energy rather than length is technically convenient: energy critical points are automatically constant-speed, and the second variation formula is cleaner. A geodesic is a local minimum of energy (and hence of length among constant-speed curves) when the second variation is positive — this fails when the geodesic passes through a conjugate point, where nearby geodesics refocus.
The Hopf-Rinow theorem connects geodesic completeness to metric completeness: on a complete Riemannian manifold, any two points are connected by a length-minimizing geodesic. This is the manifold analogue of "the shortest path between two points is a straight line." The theorem fails without completeness — on an incomplete manifold (like ℝⁿ with a point removed), geodesics can run into the missing region. The cut locus of a point p — the set of points beyond which geodesics from p cease to minimize — is a fundamental geometric object that encodes the global structure of the metric.