Editing Geodesic (section)

===Geodesic spray===
{{further|Spray (mathematics)#Geodesic}}
The geodesic flow defines a family of curves in the [[tangent bundle]]. The derivatives of these curves define a [[vector field]]  on the [[total space]] of the tangent bundle, known as the '''geodesic [[spray (mathematics)|spray]]'''.

More precisely, an affine connection gives rise to a splitting of the [[double tangent bundle]] TT''M'' into [[horizontal bundle|horizontal]] and [[vertical bundle]]s:
:<math>TTM = H\oplus V.</math>
The geodesic spray is the unique horizontal vector field ''W'' satisfying
:<math>\pi_* W_v = v\,</math>
at each point ''v''&nbsp;∈&nbsp;T''M''; here {{pi}}<sub>∗</sub>&nbsp;:&nbsp;TT''M''&nbsp;→&nbsp;T''M'' denotes the [[pushforward (differential)]] along the projection {{pi}}&nbsp;:&nbsp;T''M''&nbsp;→&nbsp;''M'' associated to the tangent bundle.

More generally, the same construction allows one to construct a vector field for any [[Ehresmann connection]] on the tangent bundle.  For the resulting vector field to be a spray (on the deleted tangent bundle T''M''&nbsp;\&nbsp;{0}) it is enough that the connection be equivariant under positive rescalings: it need not be linear.  That is, (cf. [[Ehresmann connection#Vector bundles and covariant derivatives]]) it is enough that the horizontal distribution satisfy
:<math>H_{\lambda X} = d(S_\lambda)_X H_X\,</math>
for every ''X''&nbsp;∈&nbsp;T''M''&nbsp;\&nbsp;{0} and λ&nbsp;>&nbsp;0.  Here ''d''(''S''<sub>λ</sub>) is the [[pushforward (differential)|pushforward]] along the scalar homothety <math>S_\lambda: X\mapsto \lambda X.</math>  A particular case of a non-linear connection arising in this manner is that associated to a [[Finsler manifold]].