Key Takeaway: Sprays on a manifold M can be used to study geometrically individual connections, and their universality and stability make them ideal for studying geodesic incompleteness in Riemannian manifolds.

Abstract

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.