# Category: Geodesic (mathematics)

Geodesics on an ellipsoid
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a
Geodesic bicombing
In metric geometry, a geodesic bicombing distinguishes a class of geodesics of a metric space. The study of metric spaces with distinguished geodesics traces back to the work of the mathematician Herb
Geodesic deviation
In general relativity, put another way, if two objects are set in motion along two initially parallel trajectories, the presence of a tidal gravitational force will cause the trajectories to bend towa
Theorem of the three geodesics
In differential geometry the theorem of the three geodesics, also known as Lyusternik–Schnirelmann theorem, states that every Riemannian manifold with the topology of a sphere has at least three simpl
Geodesic
In geometry, a geodesic (/ˌdʒiː.əˈdɛsɪk, -oʊ-, -ˈdiːsɪk, -zɪk/) is a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifol
Geodesic curvature
In Riemannian geometry, the geodesic curvature of a curve measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of t
Closed geodesic
In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the
Geodesics in general relativity
In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a parti
Lyusternik–Fet theorem
In mathematics, the Lyusternik–Fet theorem states that on every compact Riemannian manifold there exists a closed geodesic. It is named after Lazar Lyusternik and Abram Ilyich Fet.
Geodesic manifold
In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, you can follow a "straight" line indefinitely along any
Geodesic map
In mathematics—specifically, in differential geometry—a geodesic map (or geodesic mapping or geodesic diffeomorphism) is a function that "preserves geodesics". More precisely, given two (pseudo-)Riema
Geodesic circle
A geodesic circle is either "the locus on a surface at a constant geodesic distance from a fixed point" or a curve of constant geodesic curvature. A geodesic disk is the region on a surface bounded by
Geodesic convexity
In mathematics — specifically, in Riemannian geometry — geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds. It is common to drop the prefix "geo
Vector flow
In mathematics, the vector flow refers to a set of closely related concepts of the flow determined by a vector field. These appear in a number of different contexts, including differential topology, R
Alexandrov's uniqueness theorem
The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex
Geodesics as Hamiltonian flows
In mathematics, the geodesic equations are second-order non-linear differential equations, and are commonly presented in the form of Euler–Lagrange equations of motion. However, they can also be prese
Prime geodesic
In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime