Identity concerning harmonic maps between Riemannian manifolds
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 , RiemN Riemann curvature tensor on N and RicM Ricci curvature tensor on M . Then
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{\displaystyle {\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 }.}
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