Spray (Mathematics)
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High Quality Content by WIKIPEDIA articles! In differential geometry, a spray is a type of vector field defined on the tangent bundle of a manifold. Sprays arise naturally in Riemannian and Finsler geometry as the geodesic spray whose integral curves are precisely all geodesics as Hamiltonian flows. More generally, sprays geometrically encode quadratic quasilinear second-order ordinary differential equations on a manifold, the geodesic equation of a Riemannian or Finsler manifold being one special case of this. A spray may also be associated to any affine connection on a differentiable…mehr

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High Quality Content by WIKIPEDIA articles! In differential geometry, a spray is a type of vector field defined on the tangent bundle of a manifold. Sprays arise naturally in Riemannian and Finsler geometry as the geodesic spray whose integral curves are precisely all geodesics as Hamiltonian flows. More generally, sprays geometrically encode quadratic quasilinear second-order ordinary differential equations on a manifold, the geodesic equation of a Riemannian or Finsler manifold being one special case of this. A spray may also be associated to any affine connection on a differentiable manifold. A spray may only be defined or regular on part of the tangent bundle: the Finsler spray is defined on the deleted tangent bundle TM {0}. By contrast, sprays that are regular in a neighborhood of zero have well-behaved exponential maps associated with them, and accordingly a system of local normal coordinates around each point. Let M be a differentiable manifold. Then a spray W on M is a differentiable vector field on the tangent bundle TM (that is, a section of the double tangent bundle TTM).