# 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

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

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 |