PlanetMath: geodesic

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geodesic

(Definition)

Let be a differentiable manifold (at least two times differentiable) with affine connection $\nabla$ . The solution to the equation

$\displaystyle \nabla_{\dot\gamma}\dot{\gamma}=0$

defined in the interval

, is called a geodesic or a geodesic curve. It can be shown that if $\nabla$ is a Levi-Civita connection and

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 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 , it is also a property for a point (known as a focal point of ) where different geodesics issuing from intersects, to be the point where any given geodesic from ceases to be minimizing.

Coordinates

In coordinates the equation is given by the system

$\displaystyle \frac{d^2x_k}{dt^2}+\sum_{i,j}\Gamma^k_{ij}\frac{dx_i}{dt}\frac{dx_j}{dt}=0 \qquad 1\leq k \leq n$

where $\Gamma^k_{ij}$ is the Christoffel symbols (see entry about connection),

is the parameter of the curve and $\{x_1, \ldots , x_n\}$ are coordinates on

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

$\displaystyle \nabla_{\dot{\gamma}}{\dot{\gamma}}$	$\displaystyle = \nabla_{\sum_i\frac{dx_i}{dt}\partial_{x_i}}{\sum_j\frac{dx_j}{dt}\partial_{x_j}}$
	$\displaystyle =\sum_k\dot{\gamma}\left(\frac{dx_k}{dt}\right)\partial_{x_k}+ \sum_{i,j}\frac{dx_j}{dt}\frac{dx_i}{dt}\nabla_{\partial_{x_i}}\partial_{x_j}$
	$\displaystyle =\sum_k\left( \frac{d^2x_k}{dt^2}+\sum_{i,j} \frac{dx_i}{dt}\frac{dx_j}{dt}\Gamma^k_{ij}\right)\partial_{x_k}.$

Metric spaces

A geodesic in a metric space

is simply a continuous $f:[0,a]\to X$ such that the length of

. 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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"geodesic" is owned by Mathprof. [ full author list (4) | owner history (1) ]

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See Also: connection, connection

Also defines:

focal point, minimizing geodesic, geodesic curve

Keywords:

shortest path

Attachments:: affine parameter (Definition) by rspuzio
geodesic triangle (Definition) by Wkbj79

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Cross-references: distance, continuous, metric space, vectors, parameter, connection, Christoffel symbols, coordinates, intersects, property, infinite, antipodal point, spheres, great circles, Euclidean space, lines, points, curve, Levi-Civita connection, interval, equation, solution, affine connection, differentiable, differentiable manifold
There are 21 references to this entry.

This is version 19 of geodesic, born on 2004-01-13, modified 2007-07-20.
Object id is 5513, canonical name is Geodesic.
Accessed 5132 times total.

Classification:

AMS MSC:

53C22 (Differential geometry :: Global differential geometry :: Geodesics)

Pending Errata and Addenda

None.[ View all 6 ]

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