Formal Definitions
Let M be a differentiable manifold and (TM,πTM,M) its tangent bundle. Then a vector field H on TM (that is, a section of the double tangent bundle TTM) is a semispray on M, if any of the three following equivalent conditions holds:
- (πTM)*Hξ = ξ.
- JH=V, where J is the tangent structure on TM and V is the canonical vector field on TM\0.
- j∘H=H, where j:TTM→TTM is the canonical flip and H is seen as a mapping TM→TTM.
A semispray H on M is a (full) spray if any of the following equivalent conditions hold:
- Hλξ = λ*(λHξ), where λ*:TTM→TTM is the push-forward of the multiplication λ:TM→TM by a positive scalar λ>0.
- The Lie-derivative of H along the canonical vector field V satisfies =H.
- The integral curves t→ΦHt(ξ)∈TM\0 of H satisfy ΦHt(λξ)=ΦHλt(ξ) for any λ>0.
Let (xi,ξi) be the local coordinates on TM associated with the local coordinates (xi) on M using the coordinate basis on each tangent space. Then H is a semispray on M if and only if it has a local representation of the form
on each associated coordinate system on TM. The semispray H is a (full) spray, if and only if the spray coefficients Gi satisfy
Read more about this topic: Spray (mathematics)
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