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.