 | This article may be too technical for most readers to understand.(June 2012) |
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.
If is a smooth function, then
where is the gradient of with respect to , is the Hessian of with respect to and is the Ricci curvature tensor. [1] If is harmonic (i.e., , where is the Laplacian with respect to the metric ), 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 (by the divergence theorem) and integrating by parts the first term on the right-hand side.
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