harmonic map

A *harmonic map* is a smooth function $f \colon X \longrightarrow Y$ between two Riemannian manifolds $(X,g_X)$, $(X,g_Y)$ which is a critical point of the Dirichlet kinetic energy functional

$E(f) \coloneqq \int_{X} \Vert d f\Vert^2 dVol_X
\,.$

where $d f \in \Gamma (T^\ast X \otimes \phi^\ast T Y)$ is the derivative, where the norm is given jointly by the metrics of $X$ and $Y$ and where the volume form is that of $X$.

This is a standard kinetic action action functional for sigma models, the *Polyakov action*.

- Wikipedia,
*Harmonic map*

Discussion in the context of action functionals for theories of physics includes

- Charles Misner,
*Harmonic maps as models for physical theories*, Phys. Rev. D 18, 4510 (1978) (spire)

Discussion in the context of integrable systems includes

- Shabnam Beheshti, A. Shadi Tahvildar-Zadeh,
*Integrability and Vesture for Harmonic Maps into Symmetric Spaces*(arXiv:1209.1383)