Bochner identity - Alchetron, The Free Social Encyclopedia

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, Riem_N the Riemann curvature tensor on N and Ric_M 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 ⟩ .

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