spray space

Take a fibred manifold π:BX. Choose a vector field S over B that satisfies DπS(y)=y for the JacobianMathworldPlanetmath map Dπ:TBB over all coordinate vectors y=(y1,,yn)B. A spray field G over B is a globally defined smooth vector field associated to the first jet bundle JB1X of X that is given in local coordinates x=(x1,,xn)B as


The spray coefficients Gi(y) are second degree homogeneous functions which correspond up to nonlinear connectionsMathworldPlanetmathPlanetmath on M. Thus by Dπ the integral curves of 𝐆 must be of second order, and so given the constraints of the spray coefficients, satisfy c¨ii=2Gi(c˙). Subsequently, the pair (X,𝐆) is called a spray space.

Example 1: Choose a system of second order quasilinear ordinary differential equationsMathworldPlanetmath that satisfy


for a family of parameterized curves c, and let the system induce its corresponding spray. Then when c is also a Finsler geodesic in B with constant speed so that the covariant derivative gives DVV=0 along a vector field V, the corresponding autoparallels of the spray coefficients completely characterize a path space for B.

Title spray space
Canonical name SpraySpace
Date of creation 2013-05-03 16:21:46
Last modified on 2013-05-03 16:21:46
Owner Orphanage (1000048)
Last modified by jacou (1000048)
Numerical id 17
Author Orphanage (1000048)
Entry type Definition
Classification msc 53C60
Synonym Spray
Synonym geodesic spray
Synonym finsler spray
Defines spray spaces