Classical differential geometry and integrability of systems of hydrodynamic type

TL;DR

This paper explores the deep connections between classical differential geometry and integrable systems of hydrodynamic type, revealing new geometric insights and generalizations relevant to topological field theories.

Contribution

It uncovers the parallelism between integrable PDE systems and differential geometry, and introduces new integrable models related to N-wave resonant interactions.

Findings

Established links between integrable PDEs and differential geometry.

Derived new integrable generalizations for N-wave interactions.

Applied geometric methods to topological field theories.

Abstract

Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations is exposed. These results were recently applied by I.M.Krichever and B.A.Dubrovin to prove integrability of some models in topological field theories. Within the geometric framework we derive some new integrable (in a sense to be discussed) generalizations describing N-wave resonant interactions.