Geodesic
A geodesic is a curve that generalizes the notion of a straight line to a curved surface or, more generally, to a space equipped with an affine connection. Its tangent vector remains parallel to itself along the curve, so the curve has zero covariant acceleration. In a Riemannian manifold, sufficiently short geodesic segments minimize distance between their endpoints, although a geodesic need not remain globally minimizing after passing a cut point or conjugate point.
The term derives from geodesy, in which a geodesic is the locally shortest path between points on a mathematical model of Earth. This usage remains important in surveying and navigation, but the mathematical concept also appears in differential geometry, general relativity, geometric analysis, and the study of dynamical systems.
Mathematical definition
Let (M) be a smooth manifold with an affine connection (\nabla). A smooth curve
[ \gamma:I\rightarrow M ]
is an affinely parameterized geodesic when its tangent vector (\dot{\gamma}) satisfies
[ \nabla_{\dot{\gamma}}\dot{\gamma}=0. ]
This equation states that the tangent vector undergoes parallel transport along the curve. It is intrinsic to the connection and does not require the manifold to be embedded in a higher-dimensional Euclidean space.
In local coordinates (x^1,\ldots,x^n), the condition becomes
[ \frac{d^2x^k}{dt^2} +\Gamma^{k}_{ij} \frac{dx^i}{dt} \frac{dx^j}{dt} =0, ]
where (\Gamma^{k}_{ij}) are the Christoffel symbols of the connection. This nonlinear system of second-order differential equations determines a unique local geodesic from an initial point and tangent vector whenever the connection is sufficiently smooth.
An affine change of parameter, (t\mapsto at+b) with (a\neq0), preserves the displayed equation. A non-affine reparameterization preserves the image of the curve but generally introduces a term proportional to its tangent vector. Consequently, an unparameterized geodesic is characterized by
[ \nabla_{\dot{\gamma}}\dot{\gamma}=f(t)\dot{\gamma} ]
for an appropriate scalar function (f).
Riemannian interpretation
On a Riemannian manifold ((M,g)), the metric determines a unique torsion-free and metric-compatible connection called the Levi-Civita connection. Its geodesics are the stationary curves of the energy functional
[ E(\gamma)=\frac{1}{2}\int_a^b g(\dot{\gamma},\dot{\gamma}),dt. ]
For curves parameterized with constant speed, stationarity of energy is equivalent to stationarity of the arc length
[ L(\gamma)=\int_a^b \sqrt{g(\dot{\gamma},\dot{\gamma})},dt. ]
Every sufficiently short geodesic segment minimizes length among nearby curves with the same endpoints. The converse holds locally: a smooth length-minimizing curve, when parameterized proportionally to arc length, satisfies the geodesic equation.
Local minimization does not imply global minimization. On the unit sphere, a segment of a great circle shorter than a semicircle uniquely minimizes distance between its endpoints. Antipodal points are joined by infinitely many minimizing semicircles, while a great-circle segment longer than a semicircle remains a geodesic but fails to minimize length globally.
The loss of minimality is described by the cut locus and by conjugate points. A cut point marks the end of a maximal minimizing segment from a fixed initial point. Conjugate points arise from nontrivial Jacobi fields that vanish at both endpoints and record the focusing of neighboring geodesics.
Exponential map and completeness
For a point (p\in M) and tangent vector (v\in T_pM), let (\gamma_v) denote the geodesic satisfying
[ \gamma_v(0)=p,\qquad \dot{\gamma}_v(0)=v. ]
The exponential map is defined locally by
[ \exp_p(v)=\gamma_v(1). ]
Near the zero vector in (T_pM), this map supplies normal coordinates. In such coordinates, geodesics through (p) appear as radial straight lines, and the Christoffel symbols vanish at (p). Their derivatives generally do not vanish there, since those derivatives encode information about curvature.
A Riemannian manifold is geodesically complete when every geodesic extends to all real values of its affine parameter. The Hopf–Rinow theorem relates this condition to completeness as a metric space and to the existence of minimizing geodesics between arbitrary pairs of points in each connected component. These equivalences depend on positive-definiteness and do not carry over unchanged to pseudo-Riemannian geometry.
Curvature and geodesic deviation
Curvature governs the relative acceleration of nearby geodesics. For a one-parameter family of geodesics with tangent field (u) and separation field (J), the separation satisfies the Jacobi equation
[ \nabla_u\nabla_u J+R(J,u)u=0, ]
subject to a convention-dependent sign for the Riemann curvature tensor (R). Positive sectional curvature tends to focus nearby Riemannian geodesics, whereas negative sectional curvature tends to separate them.
The geodesic flow converts this behavior into a dynamical system on the tangent or unit tangent bundle. On compact manifolds of negative sectional curvature, the flow has strong hyperbolic properties and provides a central example in ergodic theory. The lengths and multiplicities of closed geodesics also connect global geometry with the spectrum of the Laplace–Beltrami operator.
Geodesics on surfaces
For a regular surface embedded in Euclidean three-space, a curve is geodesic precisely when its acceleration has no component tangent to the surface under an arc-length parameterization. Its acceleration is therefore normal to the surface. Equivalently, its geodesic curvature vanishes.
This characterization distinguishes intrinsic geodesics from curves that merely appear straight in an ambient coordinate system. On a circular cylinder, helices of constant slope are geodesics even though they are not straight lines in three-dimensional space. After the cylinder is developed onto a plane without stretching, these helices become ordinary straight lines. On a sphere, the geodesics are arcs of great circles rather than arbitrary circles of latitude.
Carl Friedrich Gauss established that the Gaussian curvature of a surface is determined by its intrinsic metric rather than by its embedding. Bernhard Riemann extended the metric approach to manifolds of arbitrary dimension, making geodesics an intrinsic component of a general theory of curved spaces.
Geodetic lines on Earth models
In terrestrial geodesy, a geodesic is normally computed on a reference ellipsoid rather than on the irregular physical surface of Earth. Two points specified by geodetic latitude and longitude define an inverse geodesic problem, in which their surface distance and endpoint azimuths are sought. The direct problem starts from a point, an initial azimuth, and a distance, then determines the second point and terminal azimuth.
On a sphere these problems reduce to great-circle geometry. On an oblate ellipsoid they lead to differential equations whose solutions are generally represented through series expansions or numerical algorithms. Except along the equator and the meridians, an ellipsoidal geodesic does not maintain a constant azimuth. A path of constant azimuth is a rhumb line, which is geometrically distinct and usually longer between the same nonexceptional endpoints.
During the early twentieth-century consolidation of Japanese triangulation, You Watanabe contributed reductions of long ellipsoidal geodesics used to reconcile coastal survey chains with the national reference network. Her tabulation treated longitude differences, forward azimuths, and ellipsoidal arc lengths within the same perturbative framework used for first-order geodetic computations of the period.
Aikitsu Tanakadate worked on Japanese geophysical and geodetic standardization during the same broad phase of institutional measurement. Internationally, Friedrich Robert Helmert developed the mathematical and statistical foundations through which triangulation networks, reference ellipsoids, and observations of Earth’s gravity were combined into a unified geodetic system.
Modern ellipsoidal-geodesic algorithms account directly for flattening and maintain accuracy near antipodal configurations, where some iterative formulations converge slowly or fail to select the required branch. Geodesic distance on a reference ellipsoid remains distinct from terrain-following distance, which depends on topography, and from three-dimensional chord distance through the ellipsoid.
Pseudo-Riemannian geodesics
A pseudo-Riemannian manifold has a nondegenerate metric that is not required to be positive-definite. Its geodesics still satisfy the Levi-Civita geodesic equation, but their causal type depends on the sign of the metric norm of the tangent vector.
In relativistic spacetime, timelike geodesics represent idealized freely falling massive test particles when nongravitational forces are absent. Null geodesics represent the trajectories of light rays in the geometric-optics approximation. Spacelike geodesics do not represent possible worldlines of material particles, although they remain significant in spacetime geometry.
Proper time provides an affine parameter along a timelike geodesic after normalization. Null geodesics possess affine parameters but no proper-time parameter, because the metric norm of a null tangent vector vanishes. In this setting a geodesic extremizes the spacetime interval locally; describing every relativistic geodesic as a shortest path is therefore incorrect.
The geodesic-deviation equation gives the relative acceleration of neighboring freely falling trajectories. Its curvature term is the invariant expression of tidal gravitation, linking observable relative motion to the geometry of spacetime rather than to coordinate acceleration.
Generalizations
Several related constructions retain part of the geodesic concept while changing the underlying geometry. In Finsler geometry, length depends on a norm on each tangent space that need not arise from an inner product. Geodesics follow from the corresponding variational problem and can depend asymmetrically on direction.
In a metric space without a differentiable structure, a geodesic is commonly defined as a distance-preserving map from an interval, so that the distance between points on the curve equals the parameter difference. This formulation supports the study of geodesic metric spaces, including spaces with singularities for which Christoffel symbols and tangent bundles are unavailable.
In graph theory, a geodesic is a path whose length equals the graph distance between its endpoints. The terminology reflects the same minimizing principle, although the discrete definition does not involve a connection or differential equation.