---
tag: KZJYG9DS3M
kind: concept
name: Harmonic map
msc: ["58E20", 53C43]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_map
same_as:
  wikidata: Q5659257
  wikipedia: Harmonic map
---

## Summary

A harmonic map is a smooth map between [[Riemannian manifolds|7CBT4SM83Y]] that is a critical point of the energy.
The energy is the integral of the squared size of the differential.
Harmonic maps include [[harmonic functions|0DEB0JMN0W]], [[geodesics|NSYMQMHPDN]], isometries and holomorphic maps between Kähler manifolds.
They satisfy a nonlinear elliptic system and are a central object of geometric analysis.

## Explanations

### L1 human

Think of a rubber sheet $M$ stretched over a curved surface $N$.
The energy measures how much the sheet is stretched.
A harmonic map is a position of the sheet in equilibrium, where small changes do not change the stretching to first order.
A map from a circle that runs along a great circle of a sphere at constant speed is a harmonic map.