# nLab harmonic map

### Context

#### Riemannian geometry

Riemannian geometry

## Basic definitions

• Riemannian manifold

• moduli space of Riemannian metrics

• pseudo-Riemannian manifold

• geodesic

• Levi-Civita connection

• ## Theorems

• Poincaré conjecture-theorem
• ## Applications

• gravity

• # Contents

## Idea

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.

## References

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

Last revised on November 19, 2015 at 08:59:07. See the history of this page for a list of all contributions to it.