Bochner's formula

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In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold to the Ricci curvature.

Formal statement[edit]

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 .[1] Bochner used this formula to prove the Bochner vanishing theorem.

Variations and generalizations[edit]


  1. ^ 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 .