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Affine spheres: Discretization via duality relations. (English) Zbl 0972.53012

Authors’ abstract: Affine spheres with definite and indefinite Blaschke metric are discretized in a purely geometric manner. The technique is based on simple relations between affine spheres and their duals which possess natural discrete analogues. The geometry of these duality relations is discussed in detail. Cauchy problems are posed and shown to admit unique solutions. Particular discrete definite affine spheres are shown to include regular polyhedra and some of their generalizations. Connections with integrable partial difference equations and symmetric mappings are recorded.

MSC:

53A15 Affine differential geometry
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
53A05 Surfaces in Euclidean and related spaces
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References:

[1] Blaschke W., Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie, II: Affine Differential-geometrie (1923) · JFM 49.0499.01
[2] Bobenko A., Symmetries and integrability of difference equations (Canterbury, 1996) pp 97– (1999)
[3] Bobenko A., J. Reine Angew. Math. 475 pp 187– (1996)
[4] Bobenko A., J. Differential Geom. 43 (3) pp 527– (1996)
[5] Bobenko A. I., Discrete integrable geometry and physics pp 3– (1999)
[6] Bobenko A. I., Discrete integrable geometry and physics pp 113– (1999)
[7] Bogdanov L. V., J. Phys. A 28 (5) pp L173– (1995) · Zbl 0854.35111
[8] Cie J., Phys. Lett. A 235 (5) pp 480– (1997) · Zbl 0969.37528
[9] Doliwa A., Phys. Lett. A 234 (3) pp 187– (1997) · Zbl 1044.37527
[10] Doliwa A., Phys. Lett. A 233 (4) pp 365– (1997) · Zbl 1044.37528
[11] Doliwa A., Comm. Math. Phys. 196 (1) pp 1– (1998) · Zbl 0908.35125
[12] Grammaticos B., Phys. Rev. Lett. 67 (14) pp 1825– (1991) · Zbl 0990.37518
[13] Hertrich U., Discrete integrable geometry and physics pp 59– (1999)
[14] Hirota R., J. Phys. Soc. Japan 43 (6) pp 2079– (1977) · Zbl 1334.39015
[15] Konopelchenko B. G., R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1980) pp 3075– (1998) · Zbl 1050.37034
[16] Martin R. R., The mathematics of surfaces (Manchester, 1984) pp 253– (1986)
[17] Nomizu K., Affine differential geometry (1994) · Zbl 0834.53002
[18] Nutbourne A. W., The mathematics of surfaces (Manchester, 1984) pp 233– (1986)
[19] Quispel G. R. W., Phys. Lett. A 126 (7) pp 419– (1988) · Zbl 0679.58023
[20] Quispel G. R. W., phys. D 34 (1) pp 183– (1989) · Zbl 0679.58024
[21] Sauer R., Math. Z. 52 pp 611– (1950) · Zbl 0035.37503
[22] Sauer R., Differenzengeometrie (1970)
[23] Simon U., Differential geometry: Riemannian geometry (Los Angeles, 1990) pp 585– (1993)
[24] Tzitzeica G., C. R. Acad. Sci. Paris 150 pp 955– (1910)
[25] Wunderlich W., Österreich. Akad. Wiss. Math.-Nat. Kl. S.-B. IIa. 160 pp 39– (1951)
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