# 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

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

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 ${\ddot{c}}^{ii}=2{G}^{i}(\dot{c})$. Subsequently, the pair $(X,\text{\mathbf{G}})$ is called a *spray space*.

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

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

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 |