Englert–Greenberger duality relation

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

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

Variations and generalizations

• Bochner identity
• Weitzenböck identity.

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