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</math>, </math>,


where <math> \nabla u </math> is the ] of <math>u</math> with respect to <math> g</math>. The formula is an example of a ]. Bochner used this formula to prove the ]. where <math> \nabla u </math> is the ] of <math>u</math> with respect to <math> g</math>. Bochner used this formula to prove the ].


The Bochner formula is often proved using ] or ] methods. The Bochner formula is often proved using ] or ] methods.

==Variations and generalizations==
*].


==See also== ==See also==

Revision as of 00:36, 19 October 2009

It has been suggested that this article be merged with Bochner identity. (Discuss) Proposed since February 2009.
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In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold ( M , g ) {\displaystyle (M,g)} to the Ricci curvature. More specifically, if u : M → R {\displaystyle u:M\rightarrow \mathbb {R} } is a harmonic function (i.e., △ g u = 0 {\displaystyle \triangle _{g}u=0} , where △ g {\displaystyle \triangle _{g}} is the Laplacian with respect to g {\displaystyle g} ), then

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

where ∇ u {\displaystyle \nabla u} is the gradient of u {\displaystyle u} with respect to g {\displaystyle 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

See also


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