Bochner Identity

For a smooth Harmonic Map $u:M\to N$,

$$\Delta(\vert\nabla u\vert^2)=\vert\nabla(du)\vert^2+\left\la... ...mits_N(u)(\nabla u,\nabla u)\nabla u,\nabla u}\right\rangle{},$$

where $\nabla$ is the Gradient, Ric is the Ricci Tensor, and Riem is the Riemann Tensor.

References

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