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 and is the Ricci curvature tensor. 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.
Variations and generalizations
- ^ Chow, Bennett; Lu, Peng; Ni, Lei (2006), Hamilton's Ricci flow, Graduate Studies in Mathematics, 77, Providence, RI: Science Press, New York, p. 19, ISBN 978-0-8218-4231-7, MR 2274812.