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| ==Statement of the result== | ==Statement of the result== | ||
| Let ''M'' and ''N'' be Riemannian manifolds and let ''u'' : ''M'' → ''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 | Let ''M'' and ''N'' be Riemannian manifolds and let ''u'' : ''M'' → ''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> | ||
| == |
==See also== | ||
| ⚫ | *] | ||
| ==References== | |||
| * {{cite journal | * {{cite journal | ||
| | last = Eells | | last = Eells | ||
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| ==External links== | ==External links== | ||
| * {{MathWorld|urlname=BochnerIdentity|title=Bochner identity}} | * {{MathWorld|urlname=BochnerIdentity|title=Bochner identity}} | ||
| ==See also== | |||
| ⚫ | *] | ||
| ] | ] | ||
Revision as of 17:15, 27 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
See also
References
- Eells, J (1978). "A report on harmonic maps". Bull. London Math. Soc. 10 (1): 1–68. doi:10.1112/blms/10.1.1. MR 0495450.
{{cite journal}}: Unknown parameter|coauthors=ignored (|author=suggested) (help)