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 United States mathematician Salomon Bochner.

Formal statement

If $u:M\to \mathbb{ℝ}$ is a smooth function, then

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

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

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

Bochner used this formula to prove the Bochner vanishing theorem.

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

$\underset{M}{\int }(\mathrm{\Delta }u{)}^{2}d\text{vol}=\underset{M}{\int }(|{\mathrm{\nabla }}^{2}u{|}^2+\text{Ric}(\mathrm{\nabla }u,\mathrm{\nabla }u))d\text{vol}$.

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