# Bochner formula in different forms

I am looking for a reference (better a book) that contain integral Bochner formulas for domains with boundary (I need it for 1-forms and functions only).

For example I will need the following formula: $$\int\limits_\Omega |\Delta f|^2 -|\mathrm{Hess}f|^2 +\langle\mathrm{Ric}(\nabla f),\nabla f\rangle =\int\limits_{\partial\Omega} H\cdot|\nabla f|^2,$$ where $H$ denotes mean curvature of $\partial \Omega$ and $f$ vanish on the boundary, but I will also need its analog for 1-forms and yet general boundary condition.

So far I found "Spin geometry" by Lawson and Michelsohn --- based on II/§5, it is easy to derive any formula I need (but I am sure someone did it somewhere).

• What is a relative Bochner formula? IIRC, Wu's "The Bochner Technique in Differential Geometry" (bookstore.ams.org/ctm-6) states the Bochner-Weitzenböck formulas for 1-forms and functions as corollaries. – M.G. Jun 27 '18 at 21:49
• @M.G. I mean for domains with boundary. – Anton Petrunin Jun 27 '18 at 22:15
• @M.G. No, I did not find it in this book. – Anton Petrunin Jun 28 '18 at 9:54
• Anton, it appears you are having something different in mind than what I thought initially, apologies. I was thinking of the following (stated on p.307 in Wu's book more or less): if $\psi$ is a harmonic 1-form, then $-\Delta||\psi||^2 = 2\sum_{k=1}^n||\nabla_{X_k}\psi||^2 + 2\operatorname{Ric}(\psi^\#,\psi^\#)$, where $\{X_k:1\leq k\leq n\}$ is a frame field (sorry, notations are from my own notes, so slightly different than in Wu's book). Weird, I was sure there was an explicit statement for harmonic 0-forms, but now I see there is not (but it follows from WFII). – M.G. Jun 28 '18 at 13:14
• Did you check Riemannian Geometry by Petersen? If I correctly remember he discusses the Bochner and the Bochner-Weitzenböck formulas quite extensively. – Piotr Hajlasz Jul 2 '18 at 23:35