Discrete \(\Omega\)-nets and Guichard nets via discrete Koenigs nets. (English) Zbl 07740438
Summary: We provide a convincing discretisation of Demoulin’s \(\Omega\)-surfaces along with their specialisations to Guichard and isothermic surfaces with no loss of integrable structure.
{© 2022 The Authors. Proceedings of the London Mathematical Society is copyright © London Mathematical Society.}
{© 2022 The Authors. Proceedings of the London Mathematical Society is copyright © London Mathematical Society.}
MSC:
| 53A70 | Discrete differential geometry |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 53A31 | Differential geometry of submanifolds of Möbius space |
References:
| [1] | L.Bianchi, Ricerche sulle superficie isoterme e sulla deformazione delle quadriche, Ann. Mat. Pura Appl.11 (1905), no. 1, 93-157. · JFM 36.0674.01 |
| [2] | L.Bianchi, Complementi alle ricerche sulle superficie isoterme, Ann. Mat. Pura Appl.12 (1905), 19-54. · JFM 36.0675.01 |
| [3] | L.Bianchi, Sulle superficie a rappresentazione isoterma delle linee di curvatura come inviluppi di rotolamento, Rend. Acc. Naz. Lincei24 (1915), 367-377. · JFM 45.0855.03 |
| [4] | A. I.Bobenko, U.Hertrich‐Jeromin, and I.Lukyanenko, Discrete constant mean curvature nets in space forms: Steiner’s formula and Christoffel duality, Discrete Comput. Geom.52 (2014), no. 4, 612-629. MR3279541 · Zbl 1312.53011 |
| [5] | A. I.Bobenko, H.Pottmann, and J.Wallner, A curvature theory for discrete surfaces based on mesh parallelity, Math. Ann.348 (2010), no. 1, 1-24. MR2657431 · Zbl 1219.51014 |
| [6] | A. I.Bobenko and Y. B.Suris, Isothermic surfaces in sphere geometries as Moutard nets, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.463 (2007), no. 2088, 3171-3193. MR2386657 · Zbl 1142.53014 |
| [7] | A. I.Bobenko and Y. B.Suris, Discrete differential geometry, Graduate Studies in Mathematics, vol. 98, American Mathematical Society, Providence, RI, 2008. Integrable structure. MR2467378 · Zbl 1158.53001 |
| [8] | A. I.Bobenko and Y. B.Suris, Discrete Koenigs nets and discrete isothermic surfaces, Int. Math. Res. Not. IMRN11 (2009), 1976-2012. MR2507107 · Zbl 1171.53003 |
| [9] | E.Bour, Théorie de la déformation des surfaces, J. Éc. Impériale Polytech.39 (1862), 1-148. |
| [10] | M.Brück, X.Du, J.Park, and C.‐L.Terng, The submanifold geometries associated to Grassmannian systems, Mem. Amer. Math. Soc.155 (2002), no. 735, viii+95. MR1875645 · Zbl 0998.53037 |
| [11] | F. E.Burstall, Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems, Integrable systems, geometry, and topology (C.‐L. Terng, ed.), AMS/IP Stud. Adv. Math., vol. 36, Amer. Math. Soc., Providence, RI, 2006, pp. 1-82. MR2222512 · Zbl 1105.53002 |
| [12] | F. E.Burstall, N. M.Donaldson, F.Pedit, and U.Pinkall, Isothermic submanifolds of symmetric R‐spaces, J. Reine Angew. Math.660 (2011), 191-243. MR2855825 · Zbl 1245.53049 |
| [13] | F. E.Burstall, U.Hertrich‐Jeromin, M.Pember, and W.Rossman, Polynomial conserved quantities of Lie applicable surfaces, Manuscripta Math.158 (2019), no. 3‐4, 505-546. MR3914961 · Zbl 1412.53021 |
| [14] | F. E.Burstall, U.Hertrich‐Jeromin, and W.Rossman, Discrete linear Weingarten surfaces, Nagoya Math. J.231 (2018), 55-88. MR3845588 · Zbl 1411.53007 |
| [15] | F. E.Burstall, U.Hertrich‐Jeromin, W.Rossman, and S.Santos, Discrete surfaces of constant mean curvature, Developments in differential geometry of submanifolds (S.‐P.Kobayashi (ed.), ed.), RIMS Kôkyûroku, vol. 1880, Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, pp. 133-179. |
| [16] | F. E.Burstall, U.Hertrich‐Jeromin, W.Rossman, and S.Santos, Discrete special isothermic surfaces, Geom. Dedicata174 (2015), 1-11. MR3303037 · Zbl 1308.53020 |
| [17] | P.Calapso, Sulle superficie a linee di curvatura isoterme, Rendiconti Circolo Matematico di Palermo17 (1903), 275-286. · JFM 34.0653.04 |
| [18] | T. E.Cecil, Lie sphere geometry: with applications to submanifolds, Universitext, Springer, NY, p. xii+207, 2008. MR2361414 · Zbl 1134.53001 |
| [19] | E. B.Christoffel, Ueber einige allgemeine Eigenschaften der Minimumsflächen, J. Reine Angew. Math.67 (1867), 218-228. MR1579370 · ERAM 067.1748cj |
| [20] | D.Clarke, Integrability in submanifold geometry, Ph.D. Thesis, University of Bath, 2012. MR3389373 |
| [21] | G.Darboux, Sur les surfaces isothermiques, Ann. Sci. École Norm. Sup. (3)16 (1899), 491-508. MR1508975 · JFM 30.0555.04 |
| [22] | A.Demoulin, Sur les surfaces \(R\) et les surfaces \(\Omega \), C. R. Acad. Sci. Paris153 (1911), 590-593. · JFM 42.0695.01 |
| [23] | A.Demoulin, Sur les surfaces \(R\) et les surfaces \(\Omega \), C. R. Acad. Sci. Paris153 (1911), 705-707. · JFM 42.0695.02 |
| [24] | A.Demoulin, Sur les surfaces \(\Omega \), C. R. Acad. Sci. Paris153 (1911), 927-929. · JFM 42.0634.01 |
| [25] | M.Desbrun, A. N.Hirani, M.Leok, and J. E.Marsden, Discrete exterior calculus (2005). |
| [26] | A.Doliwa, P. M.Santini, and M.Mañas, Transformations of quadrilateral lattices, J. Math. Phys.41 (2000), no. 2, 944-990. MR1737004 · Zbl 0987.37070 |
| [27] | V. V.Dolotin, A. Yu.Morozov, and Sh. R.Shakirov, The \(A_\infty \)‐structure on simplicial complexes, Teoret. Mat. Fiz.156 (2008), no. 1, 3.37. MR2488210 · Zbl 1155.81360 |
| [28] | L. P.Eisenhart, Surfaces with isothermal representation of their lines of curvature and their transformations, Trans. Amer. Math. Soc.9 (1908), no. 2, 149-177. MR1500806 · JFM 39.0684.02 |
| [29] | L. P.Eisenhart, Transformations of surfaces of Guichard and surfaces applicable to quadrics, Ann. di Mat.22 (1914), no. 3, 191-247. · JFM 45.0863.01 |
| [30] | L. P.Eisenhart, Surfaces \(\Omega\) and their transformations, Trans. Am. Math. Soc.16 (1915), no. 3, 275-310. MR1501013 · JFM 45.0861.04 |
| [31] | L. P.Eisenhart, Transformations of surfaces \(\Omega \). II, Trans. Am. Math. Soc.17 (1916), no. 1, 53-99. MR1501030 · JFM 46.1054.03 |
| [32] | D.Ferus and F.Pedit, Curved flats in symmetric spaces, Manuscripta Math.91 (1996), no. 4, 445-454. MR1421284 (97k:53074) · Zbl 0870.53043 |
| [33] | C.Guichard, Sur les surfaces isothermiques, C. R. Math. Acad. Sci. Paris130 (1900), 159-162. · JFM 31.0609.01 |
| [34] | U.Hertrich‐Jeromin, Introduction to Möbius differential geometry, London Mathematical Society Lecture Note Series, vol. 300, Cambridge University Press, Cambridge, 2003. MR2004958 · Zbl 1040.53002 |
| [35] | A. N.Hirani, Discrete exterior calculus, Ph. D. Thesis, California Institute of Technology, 2003. MR2704508 |
| [36] | P. E.Hydon and E. L.Mansfield, A variational complex for difference equations, Found. Comput. Math.4 (2004), no. 2, 187-217. MR2049870 · Zbl 1057.39013 |
| [37] | E.Musso and L.Nicolodi, The Bianchi‐Darboux transform of L‐isothermic surfaces, Internat. J. Math.11 (2000), no. 7, 911-924. MR1792958 (2002d:53012) · Zbl 0979.53010 |
| [38] | E.Musso and L.Nicolodi, Deformation and applicability of surfaces in Lie sphere geometry, Tohoku Math. J.58 (2006), no. 2, 161-187. MR2248428 · Zbl 1155.53306 |
| [39] | M.Pember, Lie applicable surfaces, Comm. Anal. Geom.28 (2020), no. 6, 1407-1450. MR4184823 · Zbl 1510.53014 |
| [40] | W. K.Schief, On the unification of classical and novel integrable surfaces. II. Difference geometry, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.459 (2003), no. 2030, 373-391. MR1997461 · Zbl 1030.53019 |
| [41] | W. K.Schief and B. G.Konopelchenko, On the unification of classical and novel integrable surfaces. I. Differential geometry, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.459 (2003), no. 2029, 67-84. MR1993345 · Zbl 1052.53010 |
| [42] | W. K.Schief, A.Szereszewski, and C.Rogers, On shell membranes of Enneper type: generalized Dupin cyclides, J. Phys. A42 (2009), no. 40, 404016-404033. MR2544280 · Zbl 1422.74066 |
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