Bochner identity

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 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