Bochner Identity

In mathematics — specifically, differential geometry — the Bochner identity is an Identity (mathematics), identity concerning harmonic maps between Riemannian manifolds. The identity is named after the United States, 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 d''u'' 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

:$\frac12 \Delta \big( , \nabla u , ^ \big) = \big, \nabla ( \mathrm u ) \big, ^ + \big\langle \mathrm_ \nabla u, \nabla u \big\rangle - \big\langle \mathrm_ (u) (\nabla u, \nabla u) \nabla u, \nabla u \big\rangle.$

See also

*Bochner's formula

References

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External links

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Differential geometry
Mathematical identities

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