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I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here:

http://www.mtm.ufsc.br/~cmdoria/Pesquisa/Seminario/artigos/DSalamon01.pdf

At the end of p. 119 Salamon considers a critical point $x$ of a Morse function $f:M\to\mathbb R$ and the set $$ N_x=\{z\mid f(\phi^T(z))\ge a-\varepsilon\}\cap\{f\le c\},$$ where $\phi$ denotes the negative gradient flow of $f$ and $c>a=f(x)$ such that there is no other critical point in $f^{-1}([a,c])$. It is ovious that $N_x$ tends to $$W^s(x)\cap\{f\le c\} $$ as $T\to\infty$, where $W^s (x)$ is the stable manifold of $f$ at $x$. But the text says more, namely that $N_x$ is a tubular neighborhood for this submanifold whose width converges to zero as $T\to\infty$.

However, I do not understand why this is true. Can anybody help me?

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  • $\begingroup$ One should be able to give an argument by looking at the normal form of the singularity, i.e., f = -x^2 + y^2 + a and the metric is Euclidean. $\endgroup$ – Guangbo Xu Nov 14 '14 at 20:51

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