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, then
where is the gradient 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.- Bochner identity
- Weitzenböck identity
