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?

  • $\begingroup$ Isn’t the integral over $\gamma$? How do you prove this? $\endgroup$ – Ivan Izmestiev Dec 2 '18 at 5:56
  • $\begingroup$ What do you mean by disk here? For topological disks the integral could be arbitrarily close to $4\pi$ no? $\endgroup$ – alesia Dec 2 '18 at 6:13
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    $\begingroup$ Also this formula can be used to prove Poincare inequalities for positively curved manifolds, but I suspect this isn't the answer you're looking for. $\endgroup$ – alesia Dec 2 '18 at 6:15
  • $\begingroup$ @alesia sorry, it is fixed now. $\endgroup$ – Anton Petrunin Dec 2 '18 at 6:42
  • $\begingroup$ @IvanIzmestiev Integral is over $\Delta_1$ --- essentially you apply Bochner formula to the restriction to $\Delta_1$ of the distance function from $\Delta_2$. $\endgroup$ – Anton Petrunin Dec 2 '18 at 6:50

I'm not sure if this is what you had in mind, but the counterpart of the Weitzenbock identity in complex geometry (apparently due to Bochnor-Kodaira-Nakano) together with Hodge theory quickly proves a number of vanishing theorems, like that the $(p,q)$ cohomology of a compact complex manifold of dimension $n$ with values in a positive holomorphic line bundle vanishes if $p + q > n$. Details and generalizations are worked out in chapter 7 of the book https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf, culminating in a proof of Kodaira's theorem about projective algebraic embeddings of complex manifolds, though that theorem is a bit more involved than just the identity.

  • $\begingroup$ I mean an application which would make you to learn Bochner's formula. Those who know homology will learn it anyway. $\endgroup$ – Anton Petrunin Dec 2 '18 at 19:25

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