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.

Proof

The Bochner formula is often proved using supersymmetry or Clifford algebra methods.

Variations and generalizations

Bochner identity
Weitzenböck identity.

References

↑ 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 .