Prove that there is no function $\phi$ on a compact manifold without boundary, such that $$\Delta\phi = c$$, where $\Delta$ is the Laplacian operator.
My intuition is that you can find a closed curve and after reparametrization, equation becomes $$\frac{\partial^2\phi}{\partial\theta^2}=c$$ on the curve, which is impossible. But I don't know if this is right or how to write it rigourously.
Since the manifold is compact, if $\Delta \phi=c$ then it is
$$0=\int_M \Delta \phi=\int_M c=c\mathrm{vol}(M).$$ Then, it follows that $c$ must be $0.$
On the other hand, any constant function $\phi$ satisfies $\Delta \phi=0.$