Bochner identity

Template:Short description 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

$\frac{1}{2}\mathrm{\Delta }(|\mathrm{\nabla }u{|}^2)=|\mathrm{\nabla }(\mathrm{d}u){|}^2+⟨{\mathrm{R}\mathrm{i}\mathrm{c}}_M\mathrm{\nabla }u,\mathrm{\nabla }u⟩-⟨{\mathrm{R}\mathrm{i}\mathrm{e}\mathrm{m}}_N(u)(\mathrm{\nabla }u,\mathrm{\nabla }u)\mathrm{\nabla }u,\mathrm{\nabla }u⟩.$

See also

• Bochner's formula

External links

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