Bochner's formula

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

Formal statement[edit]

More specifically, if  u : M \rightarrow \mathbb{R} is a harmonic function (i.e.,  \Delta_g u = 0 , where  \Delta_g is the Laplacian with respect to  g ), then

\Delta \frac{1}{2}|\nabla u| ^2 = |\nabla^2 u|^2 + \mbox{Ric}(\nabla u, \nabla u)

where  \nabla u is the gradient of u with respect to  g.[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 .