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Discrete asymptotic nets and \(W\)-congruences in Plücker line geometry. (English) Zbl 0990.37056
The author incorporates the theory of asymptotic lattices and their transformations into the theory of quadrilateral lattices. These results are direct analogs of the well-known approach to asymptotic nets in terms of conjugate nets in the Plücker quadric. He presents a direct proof of integrability of asymptotic lattices, without using the theory of quadrilateral lattices.

MSC:
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
53A05 Surfaces in Euclidean and related spaces
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[1] J. Ambjørn, J. Jurkiewicz, R. Loll, Lorentzian and Euclidean quantum gravity—analytical and numerical results, Preprint: NBI-HE-00-02, AEI-2000-0, hep-th/0001124.
[2] L. Bianchi, Lezioni di Geometria Differenziale, Zanichetti, Bologna, 1924.
[3] A. Bobenko, R. Seiler (Eds.), Discrete Integrable Geometry and Physics, Clarendon Press, Oxford, 1999.
[4] Bobenko, A.; Pinkall, U., Discrete surfaces with constant negative Gaussian curvature and the Hirota equation, J. diff. geom., 43, 527-611, (1996) · Zbl 1059.53500
[5] A. Bobenko, W.K. Schief, Discrete indefinite Affine spheres, in: A. Bobenko, R. Seiler (Eds.), Discrete Integrable Geometry and Physics, Clarendon Press, Oxford, 1999, pp. 113-138. · Zbl 0942.53013
[6] Bogdanov, L.V.; Konopelchenko, B.G., Lattice and q-difference darboux – zakharov – manakov systems via \(∂̄\) method, J. phys., 28, L173-L178, (1995) · Zbl 0854.35111
[7] P. Clarkson, F. Nijhoff (Eds.), Symmetries and Integrability of Difference Equations, Cambridge University Press, Cambridge, 1999.
[8] G. Darboux, Leçons sur la Théorie Générale des Surfaces, Vols. I-IV, Gauthier-Villars, Paris, 1887-1896.
[9] G. Darboux, Leçons sur les Systémes Orthogonaux et les Coordonnées Curvilignes, Gauthier-Villars, Paris, 1910. · JFM 41.0674.04
[10] Doliwa, A., Geometric discretisation of the Toda system, Phys. lett. A, 234, 187-192, (1997) · Zbl 1044.37527
[11] A. Doliwa, Discrete integrable geometry with ruler and compass, in: P. Clarkson, F. Nijhoff (Eds.), Symmetries and Integrability of Difference Equations, Cambridge University Press, Cambridge, 1999, pp. 122-136. · Zbl 0957.37070
[12] Doliwa, A., Quadratic reductions of quadrilateral lattices, J. geom. phys., 30, 169-186, (1999) · Zbl 0963.37061
[13] Doliwa, A.; Mañas, M.; Martı́nez Alonso, L.; Medina, E.; Santini, P.M., Multicomponent KP hierarchy and classical transformations of conjugate nets, J. phys. A, 32, 1197-1216, (1999) · Zbl 1041.81058
[14] Doliwa, A.; Santini, P.M., Multidimensional quadrilateral lattices are integrable, Phys. lett. A, 233, 365-372, (1997) · Zbl 1044.37528
[15] Doliwa, A., The symmetric, D-invariant and egorov reductions of the quadrilateral lattice, J. geom. phys., 36, 60-102, (2000) · Zbl 0997.37052
[16] Doliwa, A.; Santini, P.M.; Mañas, M., Transformations of quadrilateral lattices, J. math. phys., 41, 944-990, (2000) · Zbl 0987.37070
[17] L.P. Eisenhart, Transformations of Surfaces, Princeton University Press, Princeton, NJ, 1923. · JFM 49.0501.01
[18] Ernst, F.J., New formulation of the axially symmetric gravitational field problem, Phys. rev., 167, 1175-1178, (1968)
[19] S.P. Finikov, Theorie der Kongruenzen, Akademie Verlag, Berlin, 1959.
[20] Ganzha, E.I.; Tsarev, S.P., An algebraic formula for superposition and the completeness of the Bäcklund transformations of (2+1)-dimensional integrable systems, Russ. math. surveys, 51, 1200-1202, (1996) · Zbl 0887.35144
[21] Hirota, R., Nonlinear partial difference equations. III. discrete sine-Gordon equation, J. phys. soc. jpn., 43, 2079-2086, (1977) · Zbl 1334.39015
[22] Hirota, R., Discrete analogue of a generalized Toda equation, J. phys. soc. jpn., 50, 3785-3791, (1981)
[23] V. Hlavatý, Differential Line Geometry, Noordhoff, Groningen, 1953.
[24] S. Kauffman, L. Smolin, Combinatorial dynamics and time in quantum gravity, in: J. Kowalski-Glikman (Ed.), Towards Quantum Gravity, Springer, Berlin, 2000, pp. 101-129. · Zbl 0976.83061
[25] Klein, F., Ueber liniengeometrie und metrische geometrie, Math. ann., 5, 257-277, (1872) · JFM 04.0411.01
[26] F. Klein, Vorlesungen über Höhere Geometrie, Springer, Berlin, 1926.
[27] Konopelchenko, B.G.; Pinkall, U., Projective generalizations of lelieuvre’s formula, Geom. dedicata, 79, 81-99, (2000) · Zbl 0952.53005
[28] Konopelchenko, B.G.; Schief, W.K., Three-dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality, Proc. R. soc. London A, 454, 3075-3104, (1998) · Zbl 1050.37034
[29] V.E. Korepin, N.M. Bogoliubov, A.G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge University Press, Cambridge, 1993. · Zbl 0787.47006
[30] Krichever, I.M.; Lipan, O.; Wiegmann, P.; Zabrodin, A., Quantum integrable models and discrete classical Hirota equations, Commun. math. phys., 188, 267-304, (1997) · Zbl 0896.58035
[31] E.P. Lane, Projective Differential Geometry of Curves and Surfaces, University of Chicago Press, Chicago, IL, 1932. · JFM 58.0789.01
[32] Lelieuvre, M., Sur LES lignes asymptotiques et leur représentation sphérique, Bull. sci. math., 12, 126-128, (1888) · JFM 20.0756.01
[33] Levi, D.; Benguria, R., Bäcklund transformations and nonlinear differential-difference equations, Proc. natl. acad. sci. USA, 77, 5025-5027, (1980) · Zbl 0453.35072
[34] D. Levi, O. Ragnisco (Eds.), Side III—Symmetries and Integrability of Difference Equations, in: CMR Proceedings and Lecture Notes, Vol. 25, AMS, Providence, RI, 2000.
[35] Levi, D.; Sym, A., Integrable systems describing surfaces of non-constant curvature, Phys. lett. A, 149, 381-387, (1990)
[36] D. Levi, L. Vinet, P. Winternitz (Eds.), Symmetries and Integrability of Difference Equations, CMR Proceedings and Lecture Notes, Vol. 9, AMS, Providence, RI, 1996.
[37] Mañas, M.; Doliwa, A.; Santini, P.M., Darboux transformations for multidimensional quadrilateral lattices, part I, Phys. lett. A, 232, 99-105, (1997) · Zbl 1006.37501
[38] Moutard, Th.F., Sur la construction des équations de la forme (1/z)(∂2z/∂x∂y)=λ(x,y), qui admettenent une intégrale générale explicite, J. éc. Pol., 45, 1-11, (1878)
[39] M. Nieszporski, On discretization of asymptotic nets, University of Bialystok preprint No. IFT UwB/11/2000. · Zbl 0997.35080
[40] Nimmo, J.J.C.; Schief, W.K., Superposition principles associated with the moutard transformation. an integrable discretisation of a (2+1)-dimensional sine-Gordon system, Proc. R. soc. London A, 453, 255-279, (1997) · Zbl 0867.35086
[41] Plebański, J.F., Some solutions of complex Einstein equations, J. math. phys., 16, 2395-2402, (1975)
[42] Plücker, J., On a new geometry of space, Phil. trans. R. soc. London, 155, 725-791, (1865)
[43] R. Sauer, Differenzengeometrie, Springer, Berlin, 1970.
[44] Schief, W.K., Self-dual Einstein spaces via a permutability theorem for the tzitzeica equation, Phys. lett. A, 223, 55-62, (1996) · Zbl 1037.83503
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