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{{Merge|Bochner identity|date=October 2007}} {{Merge|Bochner identity|date=October 2007}}
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In ], '''Bochner's formula''' is a statement relating ] on a ] <math> (M, g) </math> to the ]. More specifically, if <math> u : (M, g) \rightarrow \mathbb{R} </math> is a harmonic function, so <math> \triangle_g u = 0 </math> (<math> \triangle </math> is the Laplacian operator), then <math>
In ], '''Bochner's formula''' : <math>
\triangle f = |\nabla X|^2 - Ric(X,X) \triangle \frac{1}{2}|\nabla u| ^2 = |\nabla^2 u|^2 - \mbox{Ric}(\nabla u, \nabla u)
</math>. The formula is an example of a ]. Bochner used this formula to prove the ].
<\math>, where <math> f = \frac{1}{2}g(X,X) <\math>,
is a significant result of ] in ]. Loosely speaking, it says that the difference between the two ]-like operators on the tangent bundle of a ] is a zero-order operator determined by the Ricci curvature. It is an example of a ]. 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.

Revision as of 01:17, 6 May 2008

It has been suggested that this article be merged with Bochner identity. (Discuss) Proposed since October 2007.
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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 , g ) R {\displaystyle u:(M,g)\rightarrow \mathbb {R} } is a harmonic function, so g u = 0 {\displaystyle \triangle _{g}u=0} ( {\displaystyle \triangle } is the Laplacian operator), 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)} . 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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