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Ordered Manifolds, Invariant Cone Fields, and Semigroups

A cone field assigns to each point of a differentiable manifold a cone in the tangent space. The integral curves of the cone field give rise to a local partial order which determines the cone field. It is shown that homogeneous cone fields arise äs quotients of a Lie group endowed at each point with the left translate of a given Lie wedge, where a Lie wedge is the tangent object in the Lie algebra of a local subsemigroup of the Lie group. The question of being able to extend the local partial order to a global partial order on the whole manifold is then shown to be closely tied to the question of the globality of the corresponding Lie wedge, that is, to the question of whether it is the tangent object of a (not just local) subsemigroup of the Lie group. Finally, such connections are explored in the case of a local action of the group, and it is shown that this machinery also yields analogs of Lie's Fundamental Theorems for Lie wedges and local semigroups. 1980 Mathematics Subject Classification (1985 Revision): 53C30, 22E15, 83C40. A central purpose of this paper is to expose some strikingly close connections that exist between the theory of cone fields on manifolds and the recently emerging Lie theory of semigroups (see [6]) and to derive some of the fundamental consequences. This interplay should prove fruitful in the future development of the two theories. A cone field assigns to each point of a manifold a (closed convex) cone in the tangent space of that point and may be thought of äs lying somewhere between a vector field and a distribution. We shall be particularly interested in those cone fields which are invariant under an action (perhaps only local) of a Lie group on the