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

↑ 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 .

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