Spray (mathematics) - Formal Definitions

Formal Definitions

Let M be a differentiable manifold and (TM,π_TM,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:

• (π_TM)_*H_ξ = 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:

• H_λξ = λ_*(λH_ξ), 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→Φ_Ht(ξ)∈TM\0 of H satisfy Φ_Ht(λξ)=Φ_Hλt(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