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., Δ_gu = 0, where Δ_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.