I am looking for simple (but not worn-out) application of Bochner--Weitzenböck type formulas in comparison geometry. (I want to use it as a motivation for students.)

The vanishing theorems and estimates for eigenvalues are too standard.

One of my favorite examples is the result of Fengbo Hang and Xiaodong Wang on rigidity of manifolds with boundary isometric to a unit sphere [see Rigidity Theorems for Compact Manifolds with Boundary and Positive Ricci Curvature].

By accident I found the following simpler example: Assume two discs $D$ and $D'$ with common boundary $\gamma$ bound a convex set in a positively curved three-dimensional manifold $M$. Then $\int_{D}k_1\cdot k_2$ is small if the maximal angle between the discs on $\gamma$ is small; here $k_i$ denote the principle curvatures of $D$.

Do you know more examples of that type?

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