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

\tfrac12 \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

\tfrac12 \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