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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

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    OK. Can you please explain me how should I use the compactness of each level set in order to prove this diffeomorphism? I can't figure it out. Thanks ! ]2012-02-28
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    I am sorry, I made a mistake, I removed my comment.2012-03-01
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    (Incidentally, this is a special case of Ehresmann's fibration theorem: a proper submersion of differentiable manifolds is a fiber bundle.)2012-03-31
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    I guess you can use a deformation lemma https://books.google.com.mx/books?id=8Ff2M9jBdJAC&pg=PA81&dq=deformation+lemma&hl=en&sa=X&ved=0ahUKEwjglOyxubHUAhUBLmMKHbCMA-YQ6AEIMDAC#v=onepage&q=deformation%20lemma&f=false2017-06-09

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