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 $y=(y^1,\mathrm{\dots },y^n)\in B$. Aspray field G over $B$ is a globally definedsmooth 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 differentialequations that satisfy

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

for a familyof parameterized curves $c$, and let the system induce itscorresponding spray. Then when $c$ is also a Finslergeodesic in $B$ with constant speed so that the covariantderivative gives $D_{V}V=0$ along a vector field $V$, thecorresponding autoparallels of the spray coefficients completelycharacterize a path space for $B$.