Compactified Morse trajectory spaces
Consider a smooth manifold equipped with a Morse function and a metric so that the gradient vector field satisfies the Morse-Smale conditions. Then the Morse trajectory spaces
can - under an additional technical assumption specified in  - be compactified to smooth manifolds with boundary and corners . These compactifications are constructed such that the codimension-1 strata of the boundary are given by single breaking at a critical point (except in the first case we have to add one copy of to represent trajectories of length 0),
- Introduce smooth evaluation maps ,
- Define the renormalized length by for and for all generalized (broken) Morse trajectories
- Define the metric as sum of Hausdorff distance between images and difference of renormalized lengths.
- discuss boundary&corner stratification, in particular note that (the set of trajectories with ) is isolated from all other boundary strata (made up of generalized trajectories with )