In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold to the Ricci curvature.
More specifically, if is a harmonic function (i.e., , where is the Laplacian with respect to ), then
where is the gradient of with respect to . Bochner used this formula to prove the Bochner vanishing theorem.
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.