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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 ${\displaystyle u:(M,g)\rightarrow \mathbb {R} }$ is a harmonic function, so ${\displaystyle \triangle _{g}u=0}$ (${\displaystyle \triangle }$ is the Laplacian operator), then ${\displaystyle \triangle {\frac {1}{2}}|\nabla u|^{2}=|\nabla ^{2}u|^{2}-{\mbox{Ric}}(\nabla u,\nabla u)}$. 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.

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