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

Terng, Chuu-Lian; Uhlenbeck, Karen

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].

Reviewer: V.A.Yumaguzhin (Pereslavl’-Zalesskij)

MSC:

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)

Keywords:

integrable system; triply orthogonal system; Combescure transformations; pseudospherical surface; Egorov metric; sine-Gordon equation; Bäcklund transformation; Darboux transformations