geodesic

Let $M$ be a differentiable manifold (at least two times differentiable) with affine connection $\nabla$. The solution to the equation

$${\nabla }_{\dot{\gamma }}\dot{\gamma }=0$$

defined in the interval $[0,a]$, is called a geodesic or a geodesic curve. It can be shown that if $\nabla$ is a Levi-Civita connection and $a$ is ‘small enough’, then the curve $\gamma$ is the shortest possible curve between the points $\gamma (0)$ and $\gamma (a)$, 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 (${ℝ}^n$) 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.

Coordinates

In coordinates the equation is given by the system

$$\frac{d^2{x}_k}{d{t}^2}+\sum _{i,j}{\mathrm{\Gamma }}_{ij}^k\frac{d{x}_i}{dt}\frac{d{x}_j}{dt}=0\mathit{\hspace{1em}\hspace{1em}}1\le k\le n$$

where ${\mathrm{\Gamma }}_{ij}^k$ is the Christoffel symbols (see entry about connection), $t$ is the parameter of the curve and $\{x_1,\mathrm{\dots },x_n\}$ are coordinates on $M$.

The formula follows since if $\dot{\gamma }={\displaystyle \sum _i}{\displaystyle \frac{d{x}_i}{dt}}{\partial }_{x_i}$, where $\{{\partial }_{x_1},\mathrm{\dots },{\partial }_{x_n}\}$ are the corresponding coordinate vectors, we have

${\nabla }_{\dot{\gamma }}\dot{\gamma }$ $={\nabla }_{{\sum }_i\frac{d{x}_i}{dt}{\partial }_{x_i}}{\displaystyle \sum _j}{\displaystyle \frac{d{x}_j}{dt}}{\partial }_{x_j}$

$={\displaystyle \sum _k}\dot{\gamma }\left({\displaystyle \frac{d{x}_k}{dt}}\right){\partial }_{x_k}+{\displaystyle \sum _{i,j}}{\displaystyle \frac{d{x}_j}{dt}}{\displaystyle \frac{d{x}_i}{dt}}{\nabla }_{{\partial }_{x_i}}{\partial }_{x_j}$

$={\displaystyle \sum _k}\left({\displaystyle \frac{d^2{x}_k}{d{t}^2}}+{\displaystyle \sum _{i,j}}{\displaystyle \frac{d{x}_i}{dt}}{\displaystyle \frac{d{x}_j}{dt}}{\mathrm{\Gamma }}_{ij}^k\right){\partial }_{x_k}.$

Metric spaces

A geodesic in a metric space $(X,d)$ is simply a continuous $f:[0,a]\to X$ such that the length (http://planetmath.org/LengthOfCurveInAMetricSpace) 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