A harmonic map is a smooth function $f \colon X \longrightarrow Y$ between a pair $(X,g_X)$, $(Y,g_Y)$ of Riemannian manifolds which is a critical point of the Dirichlet kinetic energy functional
where
$\mathrm{d} f \in \Gamma \big(T^\ast X \otimes \phi^\ast T Y\big)$ is the (exterior) derivative,
the norm $\Vert-\Vert$ is given jointly by the metrics of $X$ and $Y$,
the volume form $dVol$ is that of $X$.
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 immersion – Properties we have:
(harmonic equation)
A Riemannian immersion $\phi \colon \Sigma \to X$ is a harmonic map if and only if it has vanishing tension field, hence iff its second fundamental form has vanishing trace.
Original discussion:
On the history of the concept:
Review and textbooks:
Paul Baird, John C. Wood: Harmonic Morphisms Between Riemannian Manifolds, Oxford University Press (2003) [doi:10.1093/acprof:oso/9780198503620.001.0001]
Fanghua Lin, Changyou Wang: The Analysis of Harmonic Maps and Their Heat Flows, World Scientific (2008) [doi:10.1142/6679]
Frédéric Hélein, John C. Wood: Harmonic maps, in: Handbook of Global Analysis, Elsevier (2008) 417-491 [doi:10.1016/B978-044452833-9.50009-7, pdf]
Jürgen Jost, Ch. 9 in: Riemannian Geometry and Geometric Analysis, Springer (2017) [doi:10.1007/978-3-319-61860-9]
Wikipedia, Harmonic map
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.