Bochner's formula

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

Formal statement

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

• Bochner identity
• Weitzenböck identity.

References

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.