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Statement: Let $f$ be a real-valued smooth function on a compact n-manifold $M$.Suppose it have finitely many critical points $\left\{p_1,...,p_k\right\}$ with associated critical values $\left\{c_1,...,c_k\right\}$.Choose a Riemannian metric $g$ on $M$ , let $X$ be the vector field $X=\text{grad}f/|\text{grad}f|_{g}^{2}$ on $M\backslash\{p_{1},...,p_{k}\}$ , and let $\theta$ denote the flow of $X$ . Show that $f(\theta_{t}(p))=f(p)+t$ whenever $\theta_{t}(p)$ is defined.

Question: Can someone give me some hints as to how to start this problem. Thanks.

Enigma
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  • How is that vector field defined on $p_1, \cdots p_k$? –  Apr 05 '15 at 20:12
  • Sorry it should have been that the vector field is defined on $M\backslash{p_1,...,p_k}$ – Enigma Apr 05 '15 at 21:31
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    Notice your vector field is $f$-related to $d/dt$ (see my question here http://math.stackexchange.com/questions/1205489/flows-of-f-related-vector-fields ) – Pedro Apr 05 '15 at 21:33
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    Did you try to use chain rule to differentiate $f(\theta_t(p))$ with respect to $t$? –  Apr 05 '15 at 21:35
  • No I haven't tried differentiating. I'll see and get back to you if I need any help. – Enigma Apr 06 '15 at 01:29
  • Alright so I think I figured it out using John and Pedro's hint. Thanks a lot. – Enigma Apr 06 '15 at 01:52

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