Abstract:
At present, it is clear that a lot of the results of modern soliton theory can
be found in the old text-books on differential geometry and numerous old
papers of XIX century (see, for instance, [9] and references therein). So
that one of important root of soliton theory lies in differential geometry
(see, e.g. [1-7] and references therein). Itself soliton equations appeared
as the some particular cases of the Gauss-Codazzi-Mainardi equations
(GCME). At the same time, so-called integrable spin systems are some
reductions of the Gauss-Weingarten equation (GWE) [3]. In modern
soliton therminology this means that the GCME and the GWE admit
some integrable cases. On the other hand, as shown quite resently by
V.E.Zakharov in the remarkable paper [5], the (1+1)-dimensional GCME
is integrable. The aim of this notice is to consider the integrability aspects
of the GCME in 2+1 dimensions and its connections with the other
integrable equations such as the Yang-Mills-Higgs-Bogomolny equation
(YMHBE) and the self-dual Yang-Mills equation (SDYME).
