I am looking for a cool and simple application of Bochner--Weitzenböck type formulas which could be used as a motivation.

What is your favorite 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 $$\int\limits_{\Delta_1}k_1\cdot k_2\le \pi\cdot(\sin\alpha)^2,$$ where $\alpha$ is the maximal angle between the discs on $\gamma$ and $k_i$ are principle curvatures of $\Delta_1$.