Bochner's formula
In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.
Formal statement
If is a smooth function, thenwhere is the gradient of with respect to, is the Hessian of with respect to and is the Ricci curvature tensor. If is harmonic, Bochner's formula becomes
Bochner used this formula to prove the Bochner vanishing theorem.
As a corollary, if is a Riemannian manifold without boundary and is a smooth, compactly supported function, then
This immediately follows from the first identity, observing that the integral of the left-hand side vanishes and integrating by parts the first term on the right-hand side.