spray space
Take a fibred manifold π:B→X. Choose a vector field S over B
that satisfies Dπ∘S(y)=y for the Jacobian map Dπ:TB→B 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 J1BX of X that is given in local coordinates x=(x1,…,xn)∈B as
𝐆=yi∂∂xi-Gi∂∂yi. |
The spray coefficients Gi(y) are second degree homogeneous functions which correspond up to nonlinear connections 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 ¨cii=2Gi(˙c). Subsequently, the pair (X,𝐆) is called a spray space.
Example 1: Choose a system of second order quasilinear ordinary differential
equations that satisfy
¨cii+2Gi(˙c)=0 |
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 |