# 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 ($\mathbb{R}^{n}$) 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

 $\frac{d^{2}x_{k}}{dt^{2}}+\sum_{i,j}\Gamma^{k}_{ij}\frac{dx_{i}}{dt}\frac{dx_{% j}}{dt}=0\qquad 1\leq k\leq n$

where $\Gamma^{k}_{ij}$ is the Christoffel symbols (see entry about connection), $t$ is the parameter of the curve and $\{x_{1},\ldots,x_{n}\}$ are coordinates on $M$.

The formula follows since if $\displaystyle{\dot{\gamma}}=\sum_{i}\frac{dx_{i}}{dt}\partial_{x_{i}}$, where $\{\partial_{x_{1}},\ldots,\partial_{x_{n}}\}$ are the corresponding coordinate vectors, we have

 $\displaystyle\nabla_{\dot{\gamma}}{\dot{\gamma}}$ $\displaystyle=\nabla_{\sum_{i}\frac{dx_{i}}{dt}\partial_{x_{i}}}{\sum_{j}\frac% {dx_{j}}{dt}\partial_{x_{j}}}$ $\displaystyle=\sum_{k}\dot{\gamma}\left(\frac{dx_{k}}{dt}\right)\partial_{x_{k% }}+\sum_{i,j}\frac{dx_{j}}{dt}\frac{dx_{i}}{dt}\nabla_{\partial_{x_{i}}}% \partial_{x_{j}}$ $\displaystyle=\sum_{k}\left(\frac{d^{2}x_{k}}{dt^{2}}+\sum_{i,j}\frac{dx_{i}}{% dt}\frac{dx_{j}}{dt}\Gamma^{k}_{ij}\right)\partial_{x_{k}}.$

## 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 Geodesic 2013-03-22 14:06:37 2013-03-22 14:06:37 Mathprof (13753) Mathprof (13753) 22 Mathprof (13753) Definition msc 53C22 connection Connection focal point minimizing geodesic geodesic curve