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Revision as of 13:47, 2 March 2014 editDanielPie (talk | contribs)2 edits →Statement of the result: One usually has the Laplacian of the energy density which differs from the norm of the gradient by a factor of 1/2.← Previous edit Revision as of 15:04, 10 April 2014 edit undoMonkbot (talk | contribs)Bots3,695,952 editsm →References: Task 5: Fix CS1 deprecated coauthor parameter errors (bot trial)Next edit →
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| last = Eells | last = Eells
| first = J | first = J
| coauthors = Lemaire, L. |author2=Lemaire, L.
| title = A report on harmonic maps | title = A report on harmonic maps
| journal = Bull. London Math. Soc. | journal = Bull. London Math. Soc.

Revision as of 15:04, 10 April 2014

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, RiemN the Riemann curvature tensor on N and RicM the Ricci curvature tensor on M. Then

1 2 Δ ( | ∇ u | 2 ) = | ∇ ( d u ) | 2 + ⟨ R i c M ∇ u , ∇ u ⟩ − ⟨ R i e m N ( u ) ( ∇ u , ∇ u ) ∇ u , ∇ u ⟩ . {\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

References

External links

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