On the Space of Trajectories of a Generic Vector Field

TL;DR

This paper constructs a canonical compactification of the trajectory space for generic gradient-like vector fields on closed manifolds, establishing a smooth manifold with corners structure and linking differential forms to geometric complexes.

Contribution

It introduces a canonical compactification and smooth structure for the trajectory space and unstable/stable sets of generic vector fields, facilitating integration of forms and morphisms between complexes.

Findings

Constructed a canonical compactification of trajectory spaces.

Established a smooth manifold with corners structure.

Connected differential forms with geometric complexes through integration.

Abstract

This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric complex associated with a gradient like vector field and show how differential forms can be integrated on its unstable/stable sets. Integration leads to a morphism between the de Rham complex and the geometric complex.