Studying Morse theory, I am stuck on some problem.
Let $M$ be a compact smooth manifold, and $f$ is a smooth real-valued function on $M$. Choose a Riemannian metric $g$ on $M$, let $X$ be the vector field $X= \dfrac{\text{grad} f}{|\text{grad} f|_g ^2}$ on the open subset of $M$ consisting of regular points. Let $[a,b] \subseteq \mathbb{R}$ be a compact interval containing no critical values of $f$.
Let $\theta$ denote the flow of $X$. I know that $f(\theta^{(p)}(t)) = f(p)+t$ whenever $\theta^{(p)}(t)$ is defined.
I expect that $\theta^{(p)}$ is defined at least on $[0,b-a]$ if $p\in f^{-1}(a)$ . I have tried to prove this but I can't.
Could I have a proof or hints?
Thank you in advance.
The flow of a point will be well defined until it reaches a singularity of the vector field, i.e. a critical point of the function, which it will do in finite time. This is because if a maximal solution of an ode does not have $\mathbb{R}$ as a domain, it will leave every compact set in finite time. Here, because of the argument you gave, one can see that the maximal interval must always be contained in an interval of length $|\mathrm{min}(f)-\mathrm{max}(f)|$. Since $M$ is compact, it can leave every compact set only if it approximates the singularities of the vector field.