×

Curved flats and isothermic surfaces. (English) Zbl 0902.53038

Recently, J. Cieśliński, P. Goldstein and A. Sym [Phys. Lett. A 205, 37-43 (1995; Zbl 1020.53500)] determined isothermic surfaces as soliton surfaces by introducing a spectral parameter. To obtain the geometric meaning of this parameter, the autors apply the recently developed concept of curved flat [D. Ferus and E. Pedit, Manuscr. Math. 91, 445-454 (1996; Zbl 0870.53043)]. Let \(N=G/H\) be a pseudo-Riemannian symmetric space. A curved flat is a submanifold \(M\subset N\) such that the curvature operator of \(N\) vanishes on \(M\). In this paper, \(G=O_{1}^{5}\), \(K=O(3)\times O_{1}(2)\). So \(N\) is the set of spacelike 3-planes in Minkowski space \(\mathbb{R}_{1}^{5}\). Framing \(F:M\rightarrow G\) defines a sphere congruence which is the Ribacoure congruence. There is an important connection between loops of curved flats and isothermic surfaces. There is also a close connection between isothermic surfaces and solutions of Callapso’s equation. The authors illustrate this theory with the example \(f(x,y)= (r(x)\cos y,r(x) \sin y,z(x))\) such that \(r^{2}=r^{\prime 2}+z^{\prime 2}\) (surface of revolution).

MSC:

53C40 Global submanifolds
53C35 Differential geometry of symmetric spaces
PDFBibTeX XMLCite
Full Text: DOI arXiv