Bochner's formula Information & Bochner's formula Links at HealthHaven.com

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., $\triangle_g u = 0$ , where $\triangle_g$ is the Laplacian with respect to $g$ ), then

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

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.

[edit] Variations and generalizations

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