Bochner's formula

In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold $(M,\; g)$ to the Ricci curvature. More specifically, if $u\; :\; (M,\; g)\; ightarrow\; mathbb\{R\}$ is a harmonic function, so $riangle\_g\; u\; =\; 0$ ($riangle$ is the Laplacian operator), then $riangle\; frac\{1\}\{2\}|\; abla\; u|\; ^2\; =\; |\; abla^2\; u|^2\; -\; mbox\{Ric\}(\; abla\; u,\; abla\; u)$. The formula is an example of a Weitzenböck identity. Bochner used this formula to prove the Bochner vanishing theorem.

The Bochner formula is often proved using supersymmetry or Clifford algebra methods.