Bochner's formula

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

Formal statement

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

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

where ${\displaystyle \nabla u}$ is the gradient of ${\displaystyle u}$ with respect to ${\displaystyle g}$, ${\displaystyle \nabla ^{2}u}$ is the Hessian of ${\displaystyle u}$ with respect to ${\displaystyle g}$ and ${\displaystyle {\mbox{Ric}}}$ is the Ricci curvature tensor.^[1] If ${\displaystyle u}$ is harmonic (i.e., ${\displaystyle \Delta u=0}$, where ${\displaystyle \Delta =\Delta _{g}}$ is the Laplacian with respect to the metric ${\displaystyle g}$), Bochner's formula becomes

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

Bochner used this formula to prove the Bochner vanishing theorem.

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

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

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

References