Bochner identity

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.

tatement of the result

Let "M" and "N" be Riemannian manifolds and let "u" : "M" → "N" be a harmonic map. Let d denote the exterior derivative, ∇ 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

:Delta ig( | abla u |^{2} ig) = ig| abla ( mathrm{d} u ) ig|^{2} + iglangle mathrm{Ric}_{M} abla u, abla u ig angle - iglangle mathrm{Riem}_{N} (u) ( abla u, abla u) abla u, abla u ig angle.

References

* cite journal
last = Eells
first = J
coauthors = Lemaire, L.
title = A report on harmonic maps
journal = Bull. London Math. Soc.
volume = 10
year = 1978
issue = 1
pages = 1–68
issn = 0024-6093
doi = 10.1112/blms/10.1.1
MathSciNet|id=495450

External links

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