spray space

Take a fibred manifold $\pi :B\to X$. Choose a vector field $S$ over $B$ that satisfies $D\pi \circ S(y)=y$ for the Jacobian map $D\pi :TB\to B$ over all coordinate vectors $y=(y^1,\mathrm{\dots },y^n)\in B$. A spray field G over $B$ is a globally defined smooth vector field associated to the first jet bundle $J_B^{1}X$ of $X$ that is given in local coordinates $x=(x^1,\mathrm{\dots },x^n)\in B$ as

$$\text{𝐆}=y^i\frac{\partial }{\partial x^i}-G^i\frac{\partial }{\partial y^i}.$$

The spray coefficients $G^i(y)$ are second degree homogeneous functions which correspond up to nonlinear connections on $M$. Thus by $D\pi$ the integral curves of $𝐆$ must be of second order, and so given the constraints of the spray coefficients, satisfy ${\ddot{c}}^{ii}=2G^i(\dot{c})$. Subsequently, the pair $(X,\text{𝐆})$ is called a spray space.

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

$${\ddot{c}}^{ii}+2G^i(\dot{c})=0$$

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 $D_{V}V=0$ 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