Bochner's formula: Difference between revisions

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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 \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		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>

⚫	<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>,

	where <math> \nabla u </math> is the ] of <math>u</math> with respect to <math> g</math>. 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 ].

Revision as of 01:03, 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 $(M,g)$ 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.

Variations and generalizations

Weitzenböck identity.

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