Bochner identity

{{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 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

: $\frac12 \Delta \big( | \nabla u |^{2} \big) = \big| \nabla ( \mathrm{d} u ) \big|^{2} + \big\langle \mathrm{Ric}_{M} \nabla u, \nabla u \big\rangle - \big\langle \mathrm{Riem}_{N} (u) (\nabla u, \nabla u) \nabla u, \nabla u \big\rangle.$

References

{{cite journal

| last = Eells

| first = J

|author2=Lemaire, L.

| title = A report on harmonic maps

| journal = Bull. London Math. Soc.

| volume = 10

| year = 1978

| issue = 1

| pages = 1–68

| doi = 10.1112/blms/10.1.1

| mr = 495450

}}

External links

{{MathWorld|urlname=BochnerIdentity|title=Bochner identity}}

Category:Differential geometry

Category:Mathematical identities

Bochner identity

Statement of the result

See also

References

External links