Introduction (to Surveys in Differential Geometry. Vol. IV: Integrable systems). (English) Zbl 0938.35182

Terng, Chuu Lian (ed.) et al., Surveys in differential geometry. Vol. IV. A supplement to the Journal of Differential Geometry. Integral systems (integrable systems). Lectures on geometry and topology. Cambridge, MA: International Press. 5-19 (1998).
This introduction contains a short historical discussion of the early geometer’s ideas on “integrable geometric constructions”, which are prehistory of the modern theory of integrable systems. It concerns: triply orthogonal systems, the Combescure transformation, pseudospherical surfaces, Egorov metrics, sine-Gordon equations, Bäcklund transformations, and Darboux transformations of the Hill’s operators. A short review of all papers of this volume is represented, too.
For the entire collection see [Zbl 0918.00013].


35Q58 Other completely integrable PDE (MSC2000)
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
35Q53 KdV equations (Korteweg-de Vries equations)