# Compactified Morse trajectory spaces

From Polyfolds.org

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 [1] - 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),

**TODO:**

- 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 )