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# 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 distance between two arbitrary points in a manifold, is a geodesic.

Simple 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 fixed point $p$, 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 of $f$ is $a$. Of course, the length may be infinite. A geodesic metric space is a metric space where the distance between two points may be realized by a geodesic.

## Mathematics Subject Classification

53C22*no label found*

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