---
tag: 3482HSGW27
kind: definition
name: Tension field
msc: ["58E20", 53C43]
defines: W8F8S8DX5X
uses: [7CBT4SM83Y, FZBA1HA9ZM, F4NWZNTRMD, CAZYGE1T4E, E6HDE1DFXD]
sources:
  - url: https://en.wikipedia.org/wiki/Harmonic_map
---

## Definition

Let $(M, g)$ and $(N, h)$ be [[Riemannian manifolds|7CBT4SM83Y]] of dimensions $m$ and $n$, and let $u \colon M \to N$ be a smooth map.
Regard $du$ as a section of the bundle $T^*M \otimes u^* TN$ over $M$, with the connection $\nabla$ induced by the [[Levi-Civita connections|CAZYGE1T4E]] of $g$ and $h$.
The *tension field* of $u$ is the section of $u^* TN$
$$\tau(u) = \tr_g \nabla du = \sum_{i=1}^m \big( \nabla_{e_i} du \big)(e_i),$$
where $(e_i)$ is a local orthonormal frame of $TM$.
In local coordinates $x^i$ on $M$ and $y^\gamma$ on $N$, its components are
$$\tau(u)^\gamma = \Delta_g u^\gamma + \sum_{i,j,\alpha,\beta} g^{ij} \, \Gamma^\gamma_{\alpha\beta}(u) \, \frac{\partial u^\alpha}{\partial x^i} \frac{\partial u^\beta}{\partial x^j}.$$
Here $\Delta_g$ is the [[Laplace–Beltrami operator|F4NWZNTRMD]] of $g$, with the sign for which $\Delta_g = \frac{d^2}{dx^2}$ on $\R$.
The $\Gamma^\gamma_{\alpha\beta}$ are the [[Christoffel symbols|FZBA1HA9ZM]] of $h$.

## Remarks

If $M$ is compact without boundary and $u_s$ is a smooth family of maps with $u_0 = u$ and variation field $V = \partial_s u_s |_{s=0}$, then the [[energy|E6HDE1DFXD]] satisfies
$$\frac{d}{ds}\Big|_{s=0} E(u_s) = -\int_M h\big(\tau(u), V\big) \, dV_g.$$
So $-\tau(u)$ is the gradient of $E$, and $u$ is a [[harmonic map|KZJYG9DS3M]] exactly when $\tau(u) = 0$.

Example: let $M = \R$ and $\gamma \colon \R \to N$ a curve; then $\tau(\gamma) = \nabla_{\dot\gamma} \dot\gamma$, the covariant acceleration, so $\tau(\gamma) = 0$ is the geodesic equation.
For $\gamma(t) = (\cos t, \sin t, 0)$ into the unit sphere $S^2$, a great circle, $\tau(\gamma) = 0$.

Non-example: the curve $\gamma(t) = (\cos t, \sin t)$ into $\R^2$ with the Euclidean metric has $\tau(\gamma) = \ddot\gamma = (-\cos t, -\sin t) \ne 0$, so it is not harmonic.
The same parametrized circle, seen as a map into the unit circle $S^1$, has tension $0$, so the tension field depends on the target.
For $N = \R^n$ with the Euclidean metric the Christoffel symbols vanish and $\tau(u) = (\Delta_g u^1, \dots, \Delta_g u^n)$.