Misplaced Pages

This is an old revision of this page, as edited by SmackBot (talk | contribs) at 13:21, 10 October 2007 (Date/fix the maintenance tags or gen fixes). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

It has been suggested that this article be merged with Bochner's formula. (Discuss) Proposed since October 2007.

In mathematics — specifically, differential geometry — the Bochner identity is an identity concerning harmonic maps between Riemannian manifolds. The identity is named after the American mathematician Salomon Bochner.

Statement of the result

Let M and N be Riemannian manifolds and let u : M → N be a harmonic map. Let d denote the exterior derivative, ∇ the gradient, Δ the Laplace-Beltrami operator, Riem_N the Riemann curvature tensor on N and Ric_M the Ricci curvature tensor on M. Then

${\displaystyle \Delta {\big (}|\nabla u|^{2}{\big )}={\big |}\nabla (\mathrm {d} u){\big |}^{2}+{\big \langle }\mathrm {Ric} _{M}\nabla u,\nabla u{\big \rangle }-{\big \langle }\mathrm {Riem} _{N}(u)(\nabla u,\nabla u)\nabla u,\nabla u{\big \rangle }.}$

References

• Eells, J (1978). "A report on harmonic maps". Bull. London Math. Soc. 10 (1): 1–68. ISSN 0024-6093. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) MR495450

External links

• Weisstein, Eric W. "Bochner identity". MathWorld.

• Articles to be merged from October 2007
• Differential geometry
• Mathematical identities