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 TMF, 1984, Volume 60, Number 1, Pages 9–23 (Mi tmf5098)

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

M. V. Saveliev

Abstract: A method of classifying exactly and completely integrable emb.eddings in Riemannian or non-Riemannian enveloping Spaces is proposed. It is based on the algebraic approach [6, 8] to the integration of nonlinear dynamical systems. The grading conditions and the spectral composition of the Lax operators, which take values in a graded Lie algebra and distinguish the integrable classes of two-dimensional systems, are formulated in terms of the structure of the tensors of the third fundamental forms. In the framework of the method, each embedding of the three-dimensional subalgebra $sl(2)$ in a simple finite-dimensional (infinite-dimensional of finite growth) Lie algebra is associated with a definite class of exactly (completely) integrable embeddings of a two-dimensional manifold in a corresponding enveloping space equipped with the structure of .

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English version:
Theoretical and Mathematical Physics, 1984, 60:1, 638–647

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Citation: M. V. Saveliev, “Classification of exactly integrable embeddings of two-dimensional manifolds. The coefficients of the third fundamental forms”, TMF, 60:1 (1984), 9–23; Theoret. and Math. Phys., 60:1 (1984), 638–647

Citation in format AMSBIB
\Bibitem{Sav84} \by M.~V.~Saveliev \paper Classification of exactly integrable embeddings of two-dimensional manifolds. The coefficients of the third fundamental forms \jour TMF \yr 1984 \vol 60 \issue 1 \pages 9--23 \mathnet{http://mi.mathnet.ru/tmf5098} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=760437} \zmath{https://zbmath.org/?q=an:0559.53040} \transl \jour Theoret. and Math. Phys. \yr 1984 \vol 60 \issue 1 \pages 638--647 \crossref{https://doi.org/10.1007/BF01018246} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1984AAD2600002} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. E. A. Ivanov, S. O. Krivonos, “$N=4$ superextension of the Liouville equation with quaternion structure”, Theoret. and Math. Phys., 63:2 (1985), 477–486
2. M. V. Saveliev, “Multidimensional nonlinear systems”, Theoret. and Math. Phys., 69:3 (1986), 1234–1240
3. A. I. Bobenko, “Integrable surfaces”, Funct. Anal. Appl., 24:3 (1990), 227–228
4. O. I. Mokhov, “Realization of Frobenius Manifolds as Submanifolds in Pseudo-Euclidean Spaces”, Proc. Steklov Inst. Math., 267 (2009), 217–234
5. Derezin S., “Gauss–Codazzi Equations for Thin Films and Nanotubes Containing Defects”, Shell-Like Structures: Non-Classical Theories and Applications, Advanced Structured Materials, 15, ed. Altenbach H. Eremeyev V., Springer-Verlag Berlin, 2011, 531–548
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