Bochner identity
# 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 (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

    ${\frac {1}{2}}\Delta {\big|\nabla u|^{2}{\big )}={\big |}\nabla (\mathrm {d} u){\big |}^{2}+{\big \langle }\mathrm {Ric} _{M}\nabla u,\nabla u{\big \rangle }-{\big \langle }\mathrm {Riem} _{N}(u)(\nabla u,\nabla u)\nabla u,\nabla u{\big \rangle }.$
