---
tag: 5N1825CM10
kind: definition
name: Energy of a map between Riemannian manifolds
msc: ["58E20", 53C43]
defines: E6HDE1DFXD
uses: [7CBT4SM83Y, F45VQ47GPV]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_map
---

## Definition

Let $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]] of dimension $m$, possibly with boundary, and let $(N, h)$ be a Riemannian manifold of dimension $n$.
Let $u \colon M \to N$ be a smooth map.
The *energy density* of $u$ is the function $e(u) = \frac{1}{2} \abs{du}^2$ on $M$.
Here $\abs{du}^2$ is the squared norm of $du_x \colon T_x M \to T_{u(x)} N$ for the metrics $g$ and $h$.
In local coordinates $x^i$ on $M$ and $y^\alpha$ on $N$ it is
$$\abs{du}^2 = \sum_{i,j,\alpha,\beta} g^{ij}(x) \, h_{\alpha\beta}(u(x)) \, \frac{\partial u^\alpha}{\partial x^i} \frac{\partial u^\beta}{\partial x^j}.$$
The *energy* of $u$ is
$$E(u) = \int_M e(u) \, dV_g = \frac{1}{2} \int_M \abs{du}^2 \, dV_g,$$
where $dV_g$ is the Riemannian volume measure of $g$.

## Remarks

Equivalently, $\abs{du}^2 = \sum_i \abs{du(e_i)}_h^2$ for an orthonormal basis $(e_i)$ of $T_x M$, which is the trace with respect to $g$ of the pull-back metric $u^* h$.
The same formula defines $E(u)$ for maps of Sobolev class $W^{1,2}$, and for non-compact $M$ when the integral converges.
For $N = \R$ it is the Dirichlet energy $\frac12 \int_M \abs{\nabla u}^2 \, dV_g$, and for $M$ an interval it is the [[energy of a path|Q12AB1MJ64]].
When $\dim M = 2$, the energy is invariant under conformal changes of the metric $g$.
Its critical points are the [[harmonic maps|KZJYG9DS3M]].

Example: for the identity map of a compact $(M, g)$ of dimension $m$, $du(e_i) = e_i$, so $\abs{du}^2 = m$ and $E(\id) = \frac{m}{2} \operatorname{Vol}(M, g)$.

Non-example: the formula gives no finite energy for the identity map of $\R$ with the Euclidean metric, since $\R$ is not compact.
There $e = \frac{1}{2}$ everywhere, and the integral over $\R$ diverges.