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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 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 ]. In ], '''Bochner's formula''' : <math>
\triangle f = |\nabla X|^2 - Ric(X,X)
<\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 01:06, 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 : <math> \triangle f = |\nabla X|^2 - Ric(X,X) <\math>, where <math> f = \frac{1}{2}g(X,X) <\math>, is a significant result of Salomon Bochner in differential geometry. Loosely speaking, it says that the difference between the two Laplacian-like operators on the tangent bundle of a Riemannian manifold is a zero-order operator determined by the Ricci curvature. It 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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