defined in the interval , is called a geodesic or a geodesic curve. It can be shown that if is a Levi-Civita connection and is ‘small enough’, then the curve is the shortest possible curve between the points and , 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 () 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 , 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.
In coordinates the equation is given by the system
The formula follows since if , where are the corresponding coordinate vectors, we have
A geodesic in a metric space is simply a continuous such that the length (http://planetmath.org/LengthOfCurveInAMetricSpace) of is . 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.
|Date of creation||2013-03-22 14:06:37|
|Last modified on||2013-03-22 14:06:37|
|Last modified by||Mathprof (13753)|