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==Statement of the result== ==Statement of the result==


Let ''M'' and ''N'' be Riemannian manifolds and let ''u''&nbsp;:&nbsp;''M''&nbsp;→&nbsp;''N'' be a harmonic map. Let d denote the ], ∇ the ], Δ the ], Riem<sub>''N''</sub> the ] on ''N'' and Ric<sub>''M''</sub> the ] on ''M''. Then Let ''M'' and ''N'' be Riemannian manifolds and let ''u''&nbsp;:&nbsp;''M''&nbsp;→&nbsp;''N'' be a harmonic map. Let d''u'' denote the derivative (pushforward) of ''u'', ∇ the ], Δ the ], Riem<sub>''N''</sub> the ] on ''N'' and Ric<sub>''M''</sub> the ] on ''M''. Then


:<math>\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.</math> :<math>\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.</math>

Revision as of 04:27, 20 March 2012

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

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

External links

See also

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