Difference between revisions of "Compactified Morse trajectory spaces"
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Consider a smooth manifold <math>L</math> equipped with a Morse function <math>f:L\to\R</math> and a metric so that the gradient vector field <math>\nabla f</math> satisfies the Morse-Smale conditions. Then the Morse trajectory spaces | Consider a smooth manifold <math>L</math> equipped with a Morse function <math>f:L\to\R</math> and a metric so that the gradient vector field <math>\nabla f</math> satisfies the Morse-Smale conditions. Then the Morse trajectory spaces | ||
<center> | <center> | ||
Revision as of 18:31, 24 May 2017
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
![{\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}}](/images/math/a/c/3/ac36e80abe76a15538933ab31effdc2f.png)
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: smooth evaluation maps 
,
length ![\ell :\overline {\mathcal {M}}(L,L)\to [0,\infty ]](/images/math/1/9/9/1999df464cac2fa6d20df6322ce81aea.png)
given by 
for ![\gamma :[0,a]\to L](/images/math/e/2/7/e27331e5ae4baf6049fb5fd6dfbd06d7.png)
and 
for all generalized (broken) Morse trajectories 
