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Sprays, universality and stability

Abstract An important class of systems of second order differential equations can be represented as sprays on a manifold M with tangent bundle TM↠ M; that is, as certain sections of the second tangent bundle TTM ↠ TM. We consider here quadratic sprays; they correspond to symmetric linear connections on TM ↠ M and hence to principal connections on the frame bundle LM ↠ M. Such connections over M constitute a system of connections, on which there is a universal connection and through which individual connections can be studied geometrically. Correspondingly, we obtain a universal spray-like field for the system of connections and each spray on M arises as a pullback of this ‘universal spray’. The Frölicher-Nijenhuis bracket determines for each spray (or connection) a Lie subalgebra of the Lie algebra of vector fields on M and this subalgebra consists precisely of those morphisms of TTM over TM which preserve the horizontal and vertical distributions; there is a universal version of this result. Each spray induces also a Riemannian structure on LM; it isometrically embeds this manifold as a section of the space of principal connections and gives a corresponding representation of TM as a section of the space of sprays. Such embeddings allow the formulation of global criteria for properties of sprays, in a natural context. For example, if LM is incomplete in a spray-metric then it is incomplete also in the spray-metric induced by a nearby spray, because that spray induces a nearby embedding. For Riemannian manifolds, completeness of LM is equivalent to completeness of M so in the above sense we can say that geodesic incompleteness is stable; it is known to be Whitney stable.