geodesic Geodesic 2004-01-13 09:01:08 2007-07-20 14:03:51 Definition focal point minimizing geodesic geodesic curve shortest path % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \DeclareMathOperator{\Length}{Length} \PMlinkescapeword{straight} 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 \emph{geodesic} or a \emph{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 \emph{minimizing geodesic} between these points. Conversely, any curve which minimizes the \PMlinkescapetext{distance} between two arbitrary points in a manifold, is a geodesic. \PMlinkescapetext{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 \PMlinkescapetext{fixed point} $p$, it is also a property for a point $q$ (known as a \emph{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. \paragraph{Coordinates} In coordinates the equation is given by the system \[\frac{d^2x_k}{dt^2}+\sum_{i,j}\Gamma^k_{ij}\frac{dx_i}{dt}\frac{dx_j}{dt}=0 \qquad 1\leq k \leq n\] 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 \begin{align*} \nabla_{\dot{\gamma}}{\dot{\gamma}}&= \nabla_{\sum_i\frac{dx_i}{dt}\partial_{x_i}}{\sum_j\frac{dx_j}{dt}\partial_{x_j}}\\ &=\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} \\ &=\sum_k\left( \frac{d^2x_k}{dt^2}+\sum_{i,j} \frac{dx_i}{dt}\frac{dx_j}{dt}\Gamma^k_{ij}\right)\partial_{x_k}. \end{align*} \paragraph{Metric spaces} A geodesic in a metric space $(X,d)$ is simply a continuous $f:[0,a]\to X$ such that the \PMlinkname{length}{LengthOfCurveInAMetricSpace} of $f$ is $a$. Of course, the \PMlinkescapetext{length} may be infinite. A geodesic metric space is a metric space where the distance between two points may be realized by a geodesic.