Morse theory problem II

Question

I keep reading the book of Milnor, Morse theory and i have a problem. It exactly this one.

At the very end he says that is clearly that $\varphi_{b-a} $ takes $M^a$ diffeo to $M^b$ and for me is not clear enough. Can someone explain this to me?

Answer 1

The intuition should be clear: we constructed $X$ so that flowing along it corresponds precisely (at least on $[a,b]$) to moving up the level curves of $f$.

In addition to the fact (A) that $t \mapsto f(\varphi_t(q))$ is increasing at unit speed when $f(\varphi_t(q)) \in [a,b]$, note that by choosing $\rho$ suitably we can ensure that (B) $t \mapsto f(\varphi_t(q))$ is non-decreasing with speed $\le 1$ everywhere.

The upshot of (B) is that by flowing for time $b-a$ we can increase $f$ by at most $b-a$. Thus whenever $q \in M^a$ (so $f(q)\le a$), we know $f(\varphi_{b-a}(q)) \le f(q) + b-a \le b$; i.e. $\varphi_{b-a}(M^a) \subset M^b.$

On the other hand, if $p \in M^b$ (so $f(p) \le b$) then either $f(p)\le a$ and thus $f(\varphi_{a-b}(p)) \le a$, or $f(p) \in (a,b]$. In this latter case, (A) tells us that $\varphi_{a-f(p)}(p) \in M^a$ and thus $$\varphi_{a-b}(p) = \varphi_{f(p)-b} \circ\varphi_{a-f(p)}(p) \in M^a.$$ Since $\varphi_{a-b} = \varphi_{b-a}^{-1}$ this tells us $\varphi_{b-a}(M^a) \supset M^b$; so we have shown $\varphi_{b-a}(M^a) = M^b.$

Since $\varphi_t$ is the flow of some nice vector field $X$, we know that $\varphi_{b-a}$ is a diffeomorphism $M\to M$; and thus its restriction to (the submanifold-with-boundary) $M^a$ is a diffeomorphism on to the corresponding image $M^b.$