Misplaced Pages

This is an old revision of this page, as edited by Monkbot (talk | contribs) at 15:04, 10 April 2014 (→References: Task 5: Fix CS1 deprecated coauthor parameter errors (bot trial)). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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 du denote the derivative (pushforward) of u, ∇ 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

${\displaystyle {\frac {1}{2}}\Delta {\big (}|\nabla u|^{2}{\big )}={\big |}\nabla (\mathrm {d} u){\big |}^{2}+{\big \langle }\mathrm {Ric} _{M}\nabla u,\nabla u{\big \rangle }-{\big \langle }\mathrm {Riem} _{N}(u)(\nabla u,\nabla u)\nabla u,\nabla u{\big \rangle }.}$

See also

• Bochner's formula

References

• Eells, J; Lemaire, L. (1978). "A report on harmonic maps". Bull. London Math. Soc. 10 (1): 1–68. doi:10.1112/blms/10.1.1. MR 0495450.

External links

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

• Differential geometry
• Mathematical identities