Surfaces on Lie groups, on Lie algebras, and their integrability (with an appendix by Juan Carlos Alvarez Paiva). (English) Zbl 0864.53003

Fokas, A. S.; Gelfand, I. M.

Commun. Math. Phys. 177, No. 1, 203-220 (1996).

The method of the immersion of a 2-dimensional surface into a 3-dimensional Euclidean space is used to study surfaces in Lie groups and surfaces in Lie algebras. The \(n\)-dimensional generalization is also given. It is shown that a particular case of the general theory yields the characterization of an arbitrary surface in terms of \(2\times 2\) matrices. The use of Lie point groups provides an effective method for both finding as well as classifying integrable surfaces. It is proven that the integrable case of inverse harmonic \(H\) is the only case for which the Gauss-Codazzi equations admit a nontrivial Lie point group of transformations. In an appendix written by J. C. Alvarez it is shown the connection of the results obtained by the authors with Cartan’s method of moving frames.

Reviewer: G.Zet (Iaşi)

MSC:

53A05

Surfaces in Euclidean and related spaces

Keywords:

immersion; Lie point groups; integrable surfaces; Gauss-Codazzi equations

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