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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''' is a statement relating ] on a ] <math> (M, g) </math> to the ]. More specifically, if <math> u : M \rightarrow \mathbb{R} </math> is a harmonic function (i.e., <math> \triangle_g u = 0 </math>, where <math> \triangle_g </math> is the ] with respect to <math> g </math>), then
<math>
\triangle \frac{1}{2}|\nabla u| ^2 = |\nabla^2 u|^2 - \mbox{Ric}(\nabla u, \nabla u) \triangle \frac{1}{2}|\nabla u| ^2 = |\nabla^2 u|^2 - \mbox{Ric}(\nabla u, \nabla u)
</math>,
⚫ </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>. The formula 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 21:40, 23 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} . 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.

See also

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