Classification of exactly integrable embeddings of two-dimensional manifolds. The coefficients of the third fundamental forms (Q2266240)

A method is proposed for a classification of exactly and completely integrable embeddings of two-dimensional manifolds \(V_ 2\) in a Riemannian or non-Riemannian ambient space \(V_ N\). The method is constructive only for two-dimensional manifolds since there are no effective methods for integration of non-linear systems in spaces of dimension higher than two. The proposed method is based on an algebraic approach to the integration of non-linear dynamical systems associated by a Lax type representation (with curvature free ambient space) with Lie algebras G whose grading is consistent with the embeddings of the three- dimensional subalgebra sl(2) into them. The grading conditions and the spectral composition of the Lax operators with values in a graded Lie algebra, which single out integrable classes of non-linear systems are formulated in terms of the structure of the third fundamental form tensors. To each embedding of a three-dimensional subalgebra sl(2) in a simple finite-dimensional (infinite-dimensional of finite growth) Lie algebra G there corresponds a well-defined class of exactly (completely) integrable embeddings of a two-dimensional manifold in a corresponding ambient space endowed with the structure of G.