Bochner identity explained

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

(12)/(\Delta) ( | \nabla u | ^{2} ) = | \nabla ( d u ) | ^{2} + \ langle Ric _{M} \nabla u, \nabla u \ rangle - \ langle Riem _{N} (u) (\nabla u, \nabla u) \nabla u, \nabla u \ rangle.

See also

• Bochner's formula

References

• Eells. J. Lemaire, L.. A report on harmonic maps. Bull. London Math. Soc.. 10. 1978. 1. 1 - 68. 10.1112/blms/10.1.1. 495450.