Geodesic

A geodesic is a locally length-minimizing curve on a Riemannian manifold, satisfying the geodesic equation derived from the calculus of variations. In local coordinates, the equation is d²xⁱ/ds² + Γⁱ_jk (dx^j/ds)(dx^k/ds) = 0, where Γⁱ_jk are the Christoffel symbols encoding the manifold's connection.¹ This equation expresses parallel transport of the tangent vector along the curve, meaning the curve's direction does not change relative to the local geometry. Existence and uniqueness of geodesics follow from the Picard–Lindelöf theorem, guaranteeing a unique geodesic through any point in any given direction. However, global minimality is not guaranteed: a geodesic may be only locally shortest, as on a sphere where antipodal points have infinitely many great-circle geodesics of equal length.

The concept evolved from the ancient problem of finding shortest paths on the Earth's surface, tackled by Eratosthenes and later by Islamic mathematicians. In 1697, Johann Bernoulli posed the brachistochrone problem, a precursor to variational methods. Leonhard Euler's 1744 work on geodesics on surfaces laid the groundwork, and Carl Friedrich Gauss's 1827 Disquisitiones generales circa superficies curvas introduced intrinsic geometry, showing geodesics depend only on the surface's first fundamental form.² Bernhard Riemann extended these ideas to higher dimensions in his 1854 habilitation lecture, founding Riemannian geometry. The term 'geodesic' itself was coined by the physicist John Henry Poynting in 1891, though earlier mathematicians used 'geodetic line'.

In general relativity, massive objects curve spacetime, and free-falling bodies follow timelike geodesics, as described by the geodesic equation derived from the Einstein field equations.³ This explains planetary orbits and gravitational lensing. In global positioning systems (GPS), geodesics are used to compute satellite orbits and signal paths, accounting for relativistic time dilation. In computer graphics and robotics, geodesic distances on meshes enable pathfinding and texture mapping. Geodesic domes, popularized by Buckminster Fuller, are architectural structures based on geodesic lines on a sphere, though they are not mathematical geodesics in the strict sense.

Beyond the standard definition, geodesics appear in Finsler geometry, where the metric depends on direction, leading to asymmetric geodesics. In Lorentzian manifolds, null geodesics describe light paths, and closed timelike geodesics raise questions about time travel. The geodesic flow on a manifold is a Hamiltonian system, and its study connects to ergodic theory and dynamical systems. In the 19th century, Joseph Liouville found integrable geodesic flows on ellipsoids, a result with deep implications. Geodesics also play a role in medical imaging, where they model white matter fiber tracts in diffusion MRI. A notable edge case: on a torus, geodesics can be dense, never closing, illustrating the complexity of global behavior.

Riemannian manifold

A smooth manifold equipped with a positive-definite inner product on the tangent space at each point, allowing measurement of lengths and angles.

Christoffel symbols

A set of functions that describe how coordinates change along a curved manifold, used in the geodesic equation.

Great circle

The intersection of a sphere with a plane passing through its center; the shortest path between two points on a sphere.

Parallel transport

A way of moving a vector along a curve on a manifold while keeping it parallel with respect to the connection.

Null geodesic

A geodesic along which the spacetime interval is zero, representing the path of a light ray in general relativity.