Apologies in advance if this question is trivial!
Let M be a closed, connected, oriented, n-dimensional, manifold without boundary.
Let $f$ be a smooth function on M where $\Delta f \geq 0$, where $\Delta$ is the Laplacian.
From this, show that $f$ is a constant function.
What I have tried:
$\Delta f = dd^*f + d^*df$
= $dd^*f = d^*[ \sum_{i=1}^{n} \frac{\partial f}{\partial x_i} dx_i ]$
But I am not sure where to go from here, or if this is even the right direction. My end goal is to show that $Df = 0$, I am just really unsure on how to connect $\Delta f \geq 0$ to $f$ being constant. Any guidance would be much appreciated!
One way to prove this is via the maximum principle as Moishe Kohan mentions in the comments. Here is another way: Since $f$ is smooth it is bounded. Let $u = f-\inf_M f$. Then $\Delta v \geqslant 0$ and $v\geqslant 0$. It follows that $$0\leqslant \int_M v \Delta v\,d\mu_g =-\int_M \vert \nabla v \vert^2 \, d\mu_g \leqslant 0.$$ Hence, $$ \int_M \vert \nabla v \vert^2 \, d\mu_g=0,$$ so $v$ and consequently $f$ is constant.