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In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold ${\displaystyle (M,g)}$ to the Ricci curvature. More specifically, if ${\displaystyle u:M\rightarrow \mathbb {R} }$ is a harmonic function (i.e., ${\displaystyle \triangle _{g}u=0}$, where ${\displaystyle \triangle _{g}}$ is the Laplacian with respect to ${\displaystyle g}$), then

${\displaystyle \triangle {\frac {1}{2}}|\nabla u|^{2}=|\nabla ^{2}u|^{2}-{\mbox{Ric}}(\nabla u,\nabla u)}$,

where ${\displaystyle \nabla u}$ is the gradient of ${\displaystyle u}$ with respect to ${\displaystyle g}$. The formula is an example of a Weitzenböck identity. Bochner used this formula to prove the Bochner vanishing theorem.

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

See also

• Bochner identity

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