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 $u : M \rightarrow \mathbb{R}$ is a harmonic function (i.e., $\Delta_g u = 0$, where $\Delta_g$ is the Laplacian with respect to $g$), then

$\Delta \frac{1}{2}|\nabla u| ^2 = |\nabla^2 u|^2 + \mbox{Ric}(\nabla u, \nabla u)$,

where $\nabla u$ is the gradient of $u$ with respect to $g$. Bochner used this formula to prove the Bochner vanishing theorem.

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

Variations and generalizations

• Bochner identity
• Weitzenböck identity.