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 thelength (http://planetmath.org/LengthOfCurveInAMetricSpace) of $f$ is $a$.Of course, the may be infinite. A geodesic metric space is a metric spacewhere the distance between two points may be realized by a geodesic.