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.
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.
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?