Flow Lines Under Perturbation within Section Cones

We want to examine a closed smooth manifold together with a certain partial order: In the set of vector fields on , , we define a section cone - a convex subset of characterized by the property that if is a singular point for some vector field in then this is the case for all members of . We say that a point is greater than or equal to a point if there exists a flow line from to corresponding to some vector field in . The partial order that - under a certain condition - arises from the transitive closure of that relation -- gives rise to (the concept of) a di-path (directed path). That is a continuous map from the closed unit interval with the natural partial order inherited from the order of the real numbers to the manifold with the partial order defined as above, which furthermore preserves the partial orders. We examine di-paths between two critical points of minimal and of maximal index up to a particular homotopy relation.