I am looking for a cool and simple (but not worn-out) application of Bochner--Weitzenböck type formulas which could be used as a motivation.

What is your favourite example?

P.S. Here is one which I like, but want more: Assume two discs $\Delta_1$ and $\Delta_2$ with common boundary $\gamma$ bound a convex set in a positively curved three-dimensional manifold $M$. Then $\int_{\Delta_1}k_1\cdot k_2$ is small if the maximal angle between the discs on $\gamma$ is small; here $k_i$ denote the principle curvatures of $\Delta_1$.