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{{short description|Identity concerning harmonic maps between Riemannian manifolds}}
⚫ {{merge|Bochner's formula}}
In ] — specifically, ] — the '''Bochner identity''' is an ] concerning ]s between ]s. The identity is named after the ] ] ]. In ] — specifically, ] — the '''Bochner identity''' is an ] concerning ]s between ]s. The identity is named after the ] ] ].


==Statement of the result== ==Statement of the result==
⚫ Let ''M'' and ''N'' be ]s 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>\frac12 \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>
⚫ Let ''M'' and ''N'' be Riemannian manifolds and let ''u''&nbsp;:&nbsp;''M''&nbsp;&rarr;&nbsp;''N'' be a harmonic map. Let d denote the ], &nabla; the ], &Delta; the ], Riem<sub>''N''</sub> the ] on ''N'' and Ric<sub>''M''</sub> the ] on ''M''. Then


==See also==
⚫ :<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>
⚫ *]


==References== ==References==

* {{cite journal * {{cite journal
| 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.
Line 20: Line 21:
| issue = 1 | issue = 1
| pages = 1&ndash;68 | pages = 1&ndash;68
| doi = 10.1112/blms/10.1.1
| issn = 0024-6093
}} {{MathSciNet|id=495450}} | mr = 495450
}}


==External links== ==External links==

* {{MathWorld|urlname=BochnerIdentity|title=Bochner identity}} * {{MathWorld|urlname=BochnerIdentity|title=Bochner identity}}


] ]
] ]


{{differential-geometry-stub}}

Latest revision as of 19:34, 28 June 2021

Identity concerning harmonic maps between Riemannian manifolds

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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