Bochner identity
From HandWiki
Short description: 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 United States 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
- [math]\displaystyle{ \frac12 \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. }[/math]
See also
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.
External links
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