Misplaced Pages

Revision as of 01:05, 17 August 2009

It has been suggested that this article be merged with Bochner identity and Talk:Bochner identity#Merger proposal. (Discuss) Proposed since February 2009.

This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (Learn how and when to remove this message)

In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold ${\displaystyle (M,g)}$ to the Ricci curvature. More specifically, if ${\displaystyle u:M\rightarrow \mathbb {R} }$ is a harmonic function (i.e., ${\displaystyle \triangle _{g}u=0}$, where ${\displaystyle \triangle _{g}}$ is the Laplacian with respect to ${\displaystyle g}$), then

${\displaystyle \triangle {\frac {1}{2}}|\nabla u|^{2}=|\nabla ^{2}u|^{2}-{\mbox{Ric}}(\nabla u,\nabla u)}$,

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

• Bochner identity

This geometry-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Articles to be merged from February 2009
• Differential geometry
• Geometry stubs