Let M be a differentiable manifold (at least two times differentiableMathworldPlanetmathPlanetmath) with affine connectionMathworldPlanetmath . The solution to the equation


defined in the intervalMathworldPlanetmathPlanetmath [0,a], is called a geodesicMathworldPlanetmath or a geodesic curve. It can be shown that if is a Levi-Civita connectionMathworldPlanetmath and a is ‘small enough’, then the curve γ is the shortest possible curve between the points γ(0) and γ(a), and is often referred to as a minimizing geodesic between these points.

Conversely, any curve which minimizes the between two arbitrary points in a manifold, is a geodesic.

examples of geodesics includes straight lines in Euclidean space (n) and great circles on spheres (such as the equator of earth). The latter of which is not minimizing if the geodesic from the point p is extended beyond its antipodal point. This example also points out to us that between any two points there may be more than one geodesic. In fact, between a point and its antipodal point on the sphere, there are an infinite number of geodesics. Given a p, it is also a property for a point q (known as a focal point of p) where different geodesics issuing from p intersects, to be the point where any given geodesic from p ceases to be minimizing.


In coordinatesMathworldPlanetmathPlanetmath the equation is given by the system

d2xkdt2+i,jΓijkdxidtdxjdt=0  1kn

where Γijk is the Christoffel symbolsMathworldPlanetmathPlanetmath (see entry about connection), t is the parameter of the curve and {x1,,xn} are coordinates on M.

The formula follows since if γ˙=idxidtxi, where {x1,,xn} are the corresponding coordinate vectors, we have

γ˙γ˙ =idxidtxijdxjdtxj

Metric spaces

A geodesic in a metric space (X,d) is simply a continuousMathworldPlanetmath f:[0,a]X such that the length ( of f is a. Of course, the may be infinite. A geodesic metric space is a metric space where the distance between two points may be realized by a geodesic.

Title geodesic
Canonical name Geodesic
Date of creation 2013-03-22 14:06:37
Last modified on 2013-03-22 14:06:37
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 22
Author Mathprof (13753)
Entry type Definition
Classification msc 53C22
Related topic connection
Related topic Connection
Defines focal point
Defines minimizing geodesic
Defines geodesic curve