A curve on a manifold whose covariant acceleration with respect to a chosen affine connection (typically the Levi-Civita connection of a Riemannian metric) vanishes; equivalently, a curve that locally extremizes (usually minimizes) length or energy.

Principle

Geodesics are autogeodesic (connection-autoparallel) trajectories satisfying the geodesic equation (second-order ODE) and arise as Euler–Lagrange solutions of the energy functional on path space; they encode straightest possible motion compatible with the connection.

Demonstration

Great circles on the unit sphere are geodesics for the induced Levi-Civita connection: each great circle has zero covariant acceleration and locally minimizes length between nearby points on the circle until the antipodal cut point.

Misapplication

Assuming a geodesic is always the global shortest path between two points — beyond the cut locus or in presence of conjugate points a geodesic may cease to minimize length; confusing coordinate straight lines with geodesics on a curved manifold without checking the connection.

Consequence

Geodesics provide local distance-minimizing segments, define the exponential map at a point (mapping tangent vectors to points reached along geodesics), and determine normal coordinates and many curvature properties.

Reversal

Curves with nonzero covariant acceleration are non-geodesic (they feel a force or constraint); reversing the variational characterization yields curves that maximize other functionals or constrained paths like those of charged particles in electromagnetic fields.

Boundary

Requires a choice of affine connection (for metric geodesics usually the Levi-Civita connection); in metric spaces without differentiable structure 'geodesic' means shortest path but need not satisfy a differential equation; excludes discrete graphs where geodesic means combinatorial shortest path.

Semantic Tension

Tension between the differential notion (autoparallel curves solving the geodesic equation) and the metric notion (shortest paths): in Riemannian geometry they coincide locally but diverge globally; tension also with straight lines in coordinates which may be curved globally.

Synthesis

A geodesic is a curve with zero covariant acceleration for a given connection, equivalently a local extremum of length or energy, serving as the manifold's straightest or locally shortest trajectories and underpinning exponential maps and normal coordinates.