Bochner identity: Difference between revisions

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	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>\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>

	==See also==		==See also==

Revision as of 13:47, 2 March 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.

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

{\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 }.

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)