Bochner's formula

In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold $(M,g)$ to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.

Formal statement

If $u\colon M\rightarrow \mathbb {R}$ is a smooth function, then

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

where $\nabla u$ is the gradient of $u$ with respect to $g$, $\nabla ^{2}u$ is the Hessian of $u$ with respect to $g$ and ${\mbox{Ric}}$ is the Ricci curvature tensor. If $u$ is harmonic (i.e., $\Delta u=0$, where $\Delta =\Delta _{g}$ is the Laplacian with respect to the metric $g$), Bochner's formula becomes

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

Bochner used this formula to prove the Bochner vanishing theorem.

As a corollary, if $(M,g)$ is a Riemannian manifold without boundary and $u\colon M\rightarrow \mathbb {R}$ is a smooth, compactly supported function, then

$\int _{M}(\Delta u)^{2}\,d{\mbox{vol}}=\int _{M}{\Big (}|\nabla ^{2}u|^{2}+{\mbox{Ric}}(\nabla u,\nabla u){\Big )}\,d{\mbox{vol}}$.

This immediately follows from the first identity, observing that the integral of the left-hand side vanishes (by the divergence theorem) and integrating by parts the first term on the right-hand side.

Variations and generalizations

• Bochner identity
• Weitzenböck identity