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 Mat. Sb., 2000, Volume 191, Number 5, Pages 39–66 (Mi msb476)

The problem of constructive equivalence in differential geometry

B. M. Dubrov, B. P. Komrakov

International Center "Sophus Lie"

Abstract: The present paper is devoted to the algorithmic construction of diffeomorphisms establishing the equivalence of geometric structures. For structures of finite type the problem reduces to integration of completely integrable distributions with a known symmetry algebra and further to integration of Maurer–Cartan forms. We develop algorithms that reduce this problem to integration of closed 1-forms and equations of Lie type that are characterized by the fact that they admit a non-linear superposition principle. As an application we consider the problem of constructive equivalence for the structures of absolute parallelism and for the transitive Lie algebras of vector fields on manifolds.

DOI: https://doi.org/10.4213/sm476

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English version:
Sbornik: Mathematics, 2000, 191:5, 655–681

Bibliographic databases:

UDC: 514.76
MSC: Primary 53A55; Secondary 17B66, 22E60, 34A26, 53C05, 53C10, 58A10, 58A20, 5

Citation: B. M. Dubrov, B. P. Komrakov, “The problem of constructive equivalence in differential geometry”, Mat. Sb., 191:5 (2000), 39–66; Sb. Math., 191:5 (2000), 655–681

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb476
• https://doi.org/10.4213/sm476
• http://mi.mathnet.ru/eng/msb/v191/i5/p39

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Erratum

This publication is cited in the following articles:
1. Fels, ME, “Integrating scalar ordinary differential equations with symmetry revisited”, Foundations of Computational Mathematics, 7:4 (2007), 417
2. Ian. M. Anderson, Mark E. Fels, “The Cauchy Problem for Darboux Integrable Systems and Non-Linear d'Alembert Formulas”, SIGMA, 9 (2013), 017, 22 pp.
3. Peter J. Vassiliou, “Cauchy Problem for a Darboux Integrable Wave Map System and Equations of Lie Type”, SIGMA, 9 (2013), 024, 21 pp.
4. J.N.. Clelland, P.J.. Vassiliou, “A solvable string on a Lorentzian surface”, Differential Geometry and its Applications, 2013
5. Doubrov B., Kruglikov B., “On the Models of Submaximal Symmetric Rank 2 Distributions in 5D”, Differ. Geom. Appl., 35:1 (2014), 314–322
6. Catalano Ferraioli D., de Oliveira Silva L.A., “Second order evolution equations which describe pseudospherical surfaces”, J. Differ. Equ., 260:11 (2016), 8072–8108
7. Fels M.E., “on the Construction of Simply Connected Solvable Lie Groups”, J. Lie Theory, 27:1 (2017), 193–215
8. De Dona J., Tehseen N., Vassiliou P.J., “Symmetry Reduction, Contact Geometry, and Partial Feedback Linearization”, SIAM J. Control Optim., 56:1 (2018), 201–230
9. Campoamor-Stursberg R., “Reduction By Invariants and Projection of Linear Representations of Lie Algebras Applied to the Construction of Nonlinear Realizations”, J. Math. Phys., 59:3 (2018), 033502
10. Vassiliou P.J., “Cascade Linearization of Invariant Control Systems”, J. Dyn. Control Syst., 24:4 (2018), 593–623
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