Compactified Morse trajectory spaces

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Consider a smooth manifold L equipped with a Morse function f:L\to \mathbb{R} and a metric so that the gradient vector field \nabla f satisfies the Morse-Smale conditions. Then the Morse trajectory spaces

{\begin{alignedat}{4}{\mathcal  {M}}(L,L)&=\{\gamma :&[0,a]\;\;\to L&\,|\,{\dot  \gamma }=-\nabla f(\gamma ),a\geq 0\},\\{\mathcal  {M}}(p^{-},L)&=\{\gamma :&(-\infty ,0]\to L&\,|\,{\dot  \gamma }=-\nabla f(\gamma ),\lim _{{s\to -\infty }}\gamma (s)=p^{-}\},\\{\mathcal  {M}}(L,p^{+})&=\{\gamma :&[0,\infty )\;\to L&\,|\,{\dot  \gamma }=-\nabla f(\gamma ),\lim _{{s\to +\infty }}\gamma (s)=p^{+}\},\\{\mathcal  {M}}(p^{-},p^{+})&=\{\gamma :&\mathbb{R} \;\;\;\to L&\,|\,{\dot  \gamma }=-\nabla f(\gamma ),\lim _{{s\to \pm \infty }}\gamma (s)=p^{\pm }\}/\mathbb{R} .\end{alignedat}}

can - under an additional technical assumption specified in [1] - be compactified to smooth manifolds with boundary and corners {\mathcal  {M}}(\cdot ,\cdot ). 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 L to represent trajectories of length 0),

\textstyle \partial ^{1}\overline {\mathcal  {M}}(\cdot ,\cdot )\;=\;{\bigl (}\;L\cup \;{\bigr )}\;\bigsqcup _{{q\in {\text{Crit}}(f)}}\overline {\mathcal  {M}}(\cdot ,q)\times \overline {\mathcal  {M}}(q,\cdot ).


  • Introduce smooth evaluation maps {{\rm {ev}}}^{\pm }:\overline {\mathcal  {M}}(\cdot ,\cdot )\to L,
  • Define the renormalized length \ell :\overline {\mathcal  {M}}(L,L)\to [0,1] by \ell (\gamma )={\tfrac  {1}{1+a}} for \gamma :[0,a]\to L and \ell (\underline {\gamma })=1 for all generalized (broken) Morse trajectories \underline {\gamma }
  • Define the metric d_{{\overline {\mathcal  {M}}}}(\underline {\gamma },\underline {\gamma }')=d_{{{\rm {Hausdorff}}}}{\bigl (}\overline {{{\rm {im}}}(\underline {\gamma })},\overline {{{\rm {im}}}(\underline {\gamma })}{\bigr )}+{\bigl |}\ell (\underline {\gamma })-\ell (\underline {\gamma }'){\bigr |} as sum of Hausdorff distance between images and difference of renormalized lengths.
  • discuss boundary&corner stratification, in particular note that L\subset \partial ^{1}\overline {\mathcal  {M}}(L,L) (the set of trajectories with \ell =0) is isolated from all other boundary strata (made up of generalized trajectories with \ell =\infty )