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Let's suppose we have $N$ a compact Riemann manifold and a smooth function f on N. Prove that $\nabla f= 0$ at 2 or more points.

I am not very sure that this question is correct because I don't see how the fact that N is Riemannian fits.

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Awesome related fact (due to Reeb): If you can cook up such an $f$ with $\nabla f = 0$ at exactly $2$ points, $N$ is homeomorphic to a sphere! –  Jason DeVito Dec 22 '12 at 6:56
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1 Answer 1

The fact that $N$ is Riemannian is needed for the gradient to even be defined. The gradient is the metric dual to $df$, i.e. $g(\nabla f, X) = df(X)$ for all vector fields $X$ on $N$.

Hint: $f$ must attain its maximum and minimum values on $N$, since $f$ is continuous and $N$ is compact. What can you say about $\nabla f$ at a maximum/minimum?

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