spray space

Take a fibred manifold $\pi\colon B\to X$ . Choose a vector field $S$ over $B$ that satisfies $D\pi\circ S(y)=y$ for the Jacobian map $D\pi:TB\rightarrow B$ over all coordinate vectors $y=(y^{1}, \ldots , y^{n})\in B$ . 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 $x=(x^{1}, \ldots ,x^{n})\in B$ as $$\textbf{G}=y^{i}\frac{\partial}{\partial x^{i}}-G^{i}\frac{\partial}{\partial y^{i}}.$$ 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 $\mathbf{G}$ 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 $(X,{G})$ 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 $B$ .