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Bochner Identity

For a smooth harmonic map u:M->N,

Delta(|del u|^2)=|del (du)|^2+<Ric_Mdel u,del u>-<Riem_N(u)(del u,del u)del u,del u>,

where del is the gradient, Ric is the Ricci curvature tensor, and Riem is the Riemann tensor.

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References

Eels, J. and Lemaire, L. "A Report on Harmonic Maps." Bull. London Math. Soc. 10, 1-68, 1978.

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Bochner Identity

Cite this as:

Weisstein, Eric W. "Bochner Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BochnerIdentity.html

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