nLab harmonic map





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

(1)E(f)12 Xdf 2dVol X, E(f) \;\coloneqq\; \tfrac{1}{2} \textstyle{\int_{X}} \Vert \mathrm{d} f\Vert^2 dVol_X \,,


  • dfΓ(T *Xϕ *TY)\mathrm{d} f \in \Gamma \big(T^\ast X \otimes \phi^\ast T Y\big) is the (exterior) derivative,

  • the norm \Vert-\Vert is given jointly by the metrics of XX and YY,

  • the volume form dVoldVol is that of XX.

(due to Eells & Sampson 1964, §1-§2; review includes Baird & Wood 2003, §3.3; Lin & Wang 2008, Def. 1.1.1; Jost 2017, Def. 9.1.2)


The analogous formula makes sense for pseudo-Riemannian manifolds where (1) is the standard kinetic action action functional – the Polyakov action – for relativistic non-linear sigma models (such as for the relativistic particle, string or membrane, etc.)


In the terminology discussed at Riemannian immersionProperties we have:


(harmonic equation)
A Riemannian immersion ϕ:ΣX\phi \colon \Sigma \to X is a harmonic map if and only if it has vanishing tension field, hence iff its second fundamental form has vanishing trace.

(Eells & Sampson 1964 pp. 116, Baird & Wood 2003 Thm. 3.3.3)


Original discussion:

On the history of the concept:

  • Yuan-Jen Chiang, Andrea Ratto: Paying Tribute to James Eells and Joseph H. Sampson: In Commemoration of the Fiftieth Anniversary of Their Pioneering Work on Harmonic Maps, Notices of the AMS 62 4 (2015) [pdf, full issue: pdf]

Review and textbooks:

Extensive bibliography:

Discussion in the context of action functionals for theories of physics (nonlinear sigma models):

Discussion in the context of integrable systems:

Last revised on June 21, 2024 at 17:31:10. See the history of this page for a list of all contributions to it.