Bochner identity

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 the Riemann curvature tensor on N and RicM 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 ⟩ . {\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 }.}

See also

  • Bochner's formula

References

  • Eells, J; Lemaire, L. (1978). "A report on harmonic maps". Bull. London Math. Soc. 10 (1): 1–68. doi:10.1112/blms/10.1.1. MR 0495450.

External links

  • Weisstein, Eric W. "Bochner identity". MathWorld.


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