• Source: 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, 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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