Let $F:M^n \to \mathbb{R} $ be a smooth function admitting only regular values and $(M,g)$ a smooth connected riemannian manifold.
I know that the vector field $ \frac{\operatorname{grad}F}{||\operatorname{grad}F||^2} $ defined my means of the metric $g$ is a smooth one.
How can I use this fact and the flow of this vector field in order to prove that if each level set $F^{-1}(a)$ is compact, then all the nonempty level sets are diffeomorphic.
I obviouly need to use the flow $ \phi_{a,b} $ , but I'm having trouble proving that my integral curve is defined in all $\mathbb{R}$ , and that this flow is onto. Can someone help me solve this question?
Thanks in advance