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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 ! ] –  joshua Feb 28 '12 at 20:48
    
I am sorry, I made a mistake, I removed my comment. –  Thomas Rot Mar 1 '12 at 13:05
    
(Incidentally, this is a special case of Ehresmann's fibration theorem: a proper submersion of differentiable manifolds is a fiber bundle.) –  Akhil Mathew Mar 31 '12 at 7:36
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1 Answer 1

I'm sure you examine the proofs of these two theorems in [1] will help greatly in their Answer. The missing part is for completion is the proof of 1.3. End 1.2 is the dependency referere Relationships Between the continuous flow and field difereciável and I talked. The relationship $\langle \operatorname{grad} F(p), v\rangle =0 $ for all $v\in TN$ with $N=F^{-1}(a)$ is important for your Answer.

THEOREM (PROPOSITION 1.2 of [1]) Let $E$ be a Banach space and $ F $ a $ C^r,\; r\geq 0$ such that $ \pi F(\lambda,p)=p$ where $\pi:TM\to M$ is the natural projection. For every $\lambda_o \in E $ and $ p \in M $, there are neighborhoods $ W $ of $ \lambda_o $ in $E $, $ V $ of $ p $ in $ M $, a real number $ \epsilon > 0 $ and an application of class $ C^r $ $ \varphi $ such that $ \varphi (0, p,\lambda) = p $ , $ \left(\partial\backslash\partial t\right)\varphi(t,p,\lambda)=F(\varphi(t,p,\lambda),\lambda) $ for all $t\in (-\epsilon, +\epsilon)$, $p \in V$ and $\lambda\in W$. Moreover, if $ \alpha : (-\epsilon,+\epsilon) \to M $ is an integral curve of the field $ F_{\Lambda}(\cdot) = F(\lambda,\cdot) $ with $ \alpha(0)= p$, then $\alpha = \varphi_{p,\lambda}=\varphi(\cdot,p,\lambda) $.

 

THEOREM (PROPOSITION 1.3 of [1])Let $M$ be a compact manifold and a field $X\in\mathcal{X}^{r}(M)$. There is a global flow on M of class $C^r$ to $X$. This is an application $\varphi: \mathbb {R}\times M \to > M$ such that $ \varphi(0,p)=p $ and $\left(\partial\backslash\partial > t\right)\varphi(t,p)=X(\varphi(t,p)).$

PROOF. Consider any point p in M. We will show that there is an integral curve in $\mathbb{R} $, through p. Let $ (a, b) \subset \mathbb{R}$ a setting interval of an integral curve $\alpha : (a,b) \to M $ with $0\in (a,b) $ and $\alpha(0)=p$. We say that $(a,b) $ is maximal if for every interval $ J $ with the same property we have $ J \ subset (a,b) $.

We say that $ (a, b) $ is maximal, then $ b = \infty $. Otherwise, consider a sequence $ t_n\to b $, $t_n\in(a,b) $. Since $ M $ is compact, we can assume, passing through a subsequence if necessary, which $\alpha(t_n) $ converges to $ q \in M $. Let $\varphi(-\epsilon,+\epsilon)\times T_q \to M$ a local flow of $X$ in $q$. Take $n_o$ such that $b-t_{n_o}<\epsilon\backslash 2$ and $\alpha(t_{n_o})\in V_q$. Let $\gamma : (a, t_{n_o}+\epsilon) \to M$, with $\gamma(t)=\alpha(t)$, if $t\leq$, and $\gamma(t-t_{n_o},\alpha(t_{n_o}))$ if $t\geq t_{n_o}$. Then $\gamma$ is a integral curve of $X$, that is a contradition because $(a,b]\subset(a,t_{n_o})$. Similarly it is shown that $a=-\infty$. Therefore there is integral curve with $\alpha : \mathbb{R} \to M$ with $\alpha(0)=p$. By uniquiness $\alpha$ is unique.

We define $\varphi(t,p)=\alpha(t) $. Of course, $\varphi(o,p)=p $ and $\left(\partial\backslash\partial t\right)\varphi(t,p)=X(\varphi(t,p,\lambda))$. Let us show that $\varphi(t+s,p)=\varphi(t,\varphi(s,p)) $ for $ t, s \in \mathbb{R}$ and $ p\in M $. Let $\beta(t)\varphi(t+s,p)$ and $\gamma(t)=\varphi(t,\varphi(s,p))$. We have $ \beta $ and $ \alpha $ are integral curves of $ X $ and $\beta(0)=\gamma(0)=\varphi(s,p) $, which proves the assertion. Finally we show that $ \varphi $ is of class $ C^r $. Let $ p \in M $ $ $ and a local flow of $ X $, which is of class $ C^r $ by Proposition 1.2. Also the uniqueness of the solutions $ \psi $ is the restriction of $ \varphi $ a $ (-\epsilon_p,+\epsilon_p)\times V_p$. In particular, $\varphi_t=\varphi(t,\cdot ) $ is of class $ C^r $ in $ V_p $ in $ M $ for $ | t | <\epsilon $. Moreover for any $ t \in \mathbb{R} $, taking $ | t/n| <\epsilon $ for some integer $ n $, we have $\varphi_t=\varphi_{t/n}\circ\cdots\circ \varphi_{t/n}$ is of class $ C^r $in a neighborhood of $ (t_o, p_o) $. In fact, for $|t-t_o|<\epsilon_{p_o}$ end $p\in V_{p_o} $ and we have $\varphi(t,p)=\varphi_{t_o}\circ\varphi(t-t_o,p)$ is of class $C^r $, since $\varphi_{t_o}$ and $\varphi_{(-\epsilon_{p_o},+\epsilon_{p_o})\times V_{p_o}} $ and are of class $ C^r $. And this completes the proof.

[1] Palis, Jacob and Melo, Welington de. Geometric theory of dynamical systems: an introduction.

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Thanks a lot ! !!!!!!!!! –  joshua Mar 1 '12 at 8:55
    
Thank Jacob Palis and Welington de Melo for writing a great book with fundamental results as the Global Flow Theorem on manifolds. This theorem is on page 11 of Chapter 1 of this book. Now just use the idea of the answer that I deleted or some other basic idea. –  Elias Mar 1 '12 at 11:07
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AnalysisDude, you can accept this answer by clicking on the check mark (near where you vote for this answer). –  The Chaz 2.0 Mar 23 '12 at 1:22
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