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| Title of object: |
geodesic |
| Canonical Name: |
Geodesic |
| Type: |
Definition |
| Created on: |
2004-01-13 09:01:08 |
| Modified on: |
2007-07-20 12:56:16 |
| Classification: |
msc:53C22 |
| Keywords: |
shortest path |
| Defines: |
focal point, minimizing geodesic, geodesic curve |
Preamble:
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Content:
\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
length of $f$ is $a$. Length of $f$ is defined as follows:
$$
\Length(f) = \sup\left\{\sum_{i=1}^n d(f(x_{i-1},f(x_i)) \mid 0=x_0<x_1<\cdots<x_n=a\right\}.
$$
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.
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