Bochner identity

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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.

[edit] Statement of the result

Let M and N be Riemannian manifolds and let u : M → N be a harmonic map. Let d denote the exterior derivative, ∇ 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

$\Delta \big( | \nabla u |^{2} \big) = \big| \nabla ( \mathrm{d} u ) \big|^{2} + \big\langle \mathrm{Ric}_{M} \nabla u, \nabla u \big\rangle - \big\langle \mathrm{Riem}_{N} (u) (\nabla u, \nabla u) \nabla u, \nabla u \big\rangle.$