Harmonic maps in unfashionable geometries
نویسندگان
چکیده
Many topics in integrable surface geometry may be unified by application of the highly developed theory of harmonic maps of surfaces into (pseudo-)Riemannian symmetric spaces. On the one hand, such harmonic maps comprise an integrable system with spectral deformations, algebro-geometric solutions and dressing actions of loop groups generated by Bäcklund transforms [5], [6], [14], [21], [24]. On the other hand, several integrable classes of surface are characterized by harmonicity of a suitable Gauss map. Thus, a surface f : M → R3 has constant mean curvature H if and only if its Gauss map M → S is harmonic. Again, such a surface has constant Gauss curvature K if and only if its Gauss map is harmonic with respect to the metric on M provided by the second fundamental form of f. The theory of harmonic maps now provides a conceptual explanation of the classical integrable aspects of such surfaces such as associated families, Lie and Bäcklund transformations.
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