Classical Notions and Problems in Thurston Geometries

Yıl 2023, Cilt: 16 Sayı: 2, 608 - 643, 29.10.2023

Jenő Szirmai

https://doi.org/10.36890/iejg.1221802

Cited By: 1

Öz

Of the Thurston geometries, those with constant curvature geometries (Euclidean $ E^3$, hyperbolic $ H^3$, spherical $ S^3$) have been extensively studied, but the other five geometries, $ H^2\times R$, $ S^2\times R$, $Nil$, $\widetilde{SL_2 R}$, $Sol$ have been thoroughly studied only from a differential geometry and topological point of view. However, classical concepts highlighting the beauty and underlying structure of these geometries -- such as geodesic curves and spheres, the lattices, the geodesic triangles and their surfaces, their interior sum of angles and similar statements to those known in constant curvature geometries -- can be formulated. These have not been the focus of attention. In this survey, we summarize our results on this topic and pose additional open questions.

Anahtar Kelimeler

Thurston geometries, geodesic curves, geodesic triangles, spheres, sphere packings and coverings, lattices

Kaynakça

• [1] Bezdek, K.: Sphere Packings Revisited. Eur. J. Combin. 27/6, 864–883 (2006).
• [2] Böröczky, K.: Packing of spheres in spaces of constant curvature. Acta Math. Acad. Sci. Hungar. 32, 243-261 (1978).
• [3] Brodaczewska, K.: Elementargeometrie in Nil. Dissertation (Dr. rer. nat.) Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden (2014).
• [4] Bölcskei, A., Szilágyi, B.: Frenet Formulas and Geodesics in Sol Geometry. Beitr. Algebra Geom. 48/2, 411-421 (2007).
• [5] Chavel, I.: Riemannian Geometry: A Modern Introduction. Cambridge Studies in Advances Mathematics, Cambridge (2006).
• [6] Cheeger, J., Ebin, D.G.: Comparison Theorems in Riemannian Geometry. American Mathematical Society. (2006).
• [7] Csima, G., Szirmai, J.: Interior angle sum of translation and geodesic triangles in^SL2R space. Filomat. 32/14, 5023–5036 (2018).
• [8] Cavichioli, A., Molnár, E., Spaggiari, F., Szirmai, J.: Some tetrahedron manifolds with Sol geometry. J. Geometry. 105/3, 601-614 (2014).
• [9] Divjak, B., Erjavec, Z., Szabolcs, B., Szilágyi, B.: Geodesics and geodesic spheres in^SL2R geometry. Math. Commun. 14/2, 413-424 (2009).
• [10] Erjavec, Z., Horvat, D.: Biharmonic curves in^SL2R space. Math. Commun. 19 (2), 291-299 (2014).
• [11] Erjavec, Z.: Minimal surfaces in^SL2R space. Glas. Mat. Ser. III. 50, 207-221 (2015).
• [12] Erjavec, Z.: On Killing magnetic curves in^SL2R geometry. Rep. Math. Phys. 84 (3), 333-350 (2019).
• [13] Erjavec, Z.: On a certain class of Weingarten surfaces in Sol space. Int. J. Appl. Math. 28 (5), 507-514 (2015).
• [14] Erjavec, Z., Inoguchi, J.: On magnetic curves in almost cosymplectic Sol space. Results Math. 75:113, 16 pg (2020).
• [15] Erjavec, Z., Inoguchi, J.: Killing magnetic curves in Sol space. Math. Phys. Anal. Geom., 21:15, 15 pg (2018).
• [16] Eper, M., Szirmai, J.: Coverings with congruent and non-congruent hyperballs generated by doubly truncated Coxeter orthoschemes. Contributions to Discrete Mathematics. 17 No.2, 23-40 (2022), https://doi.org/10.11575/cdm.v17i2.
• [17] Farkas, Z. J.: The classification of S2×R space groups. Beitr. Algebra Geom. 42, 235-250 (2001).
• [18] Fejes Tóth, G., Kuperberg, W.: Packing and Covering with Convex Sets, Handbook of Convex Geometry Volume B, eds. Gruber, P.M., Willis J.M., pp. 799-860, North-Holland, (1983).
• [19] Fejes Tóth, G., Kuperberg, G., Kuperberg, W.: Highly Saturated Packings and Reduced Coverings. Monatsh. Math. 125/2, 127-145 (1998).
• [20] Fejes Tóth, L.: Regular Figures, Macmillan New York, 1964.
• [21] Hales, T. C.: Historical Overview of the Kepler Conjecture. Discrete and Computational Geometry, 35, 5-20 (2006).
• [22] Inoguchi, J.: Minimal translation surfaces in the Heisenberg group Nil3. Geom. Dedicata 161/1, 221-231 (2012).
• [23] Kobayashi, S., Nomizu, K.: Fundation of differential geometry, I. Interscience, Wiley, New York (1963).
• [24] Kozma, R. T., Szirmai, J.: Optimally dense packings for fully asymptotic Coxeter tilings by horoballs of different types. Monatsh. Math. 168/1, 27-47 (2012).
• [25] Kozma, R. T., Szirmai, J.: New Lower Bound for the Optimal Ball Packing Density of Hyperbolic 4-space. Discrete Comput. Geom. 53/1, 182-198 (2015), https://doi.org/10.1007/s00454-014-9634-1.
• [26] Kozma, R. T., Szirmai, J.: New horoball packing density lower bound in hyperbolic 5-space. Geom. Dedicata. 206/1, 1-25 (2020), https://doi.org/10.1007/s10711-019-00473-x.
• [27] Kozma, R. T., Szirmai, J.: Horoball Packing Density Lower Bounds in Higher Dimensional Hyperbolic n-space for 6 ≤ n ≤ 9. Geom. Dedicata. (2023), https://doi.org/10.1007/s10711-023-00779-x.
• [28] Kurusa, Á.: Ceva’s and Menelaus’ theorems in projective-metric spaces. J. Geom. 110/2, (2019), https://doi.org/10.1007/s00022-019-0495-x.
• [29] Manzano, M.J., Torralbo, F.: New Examples of Constant Mean Curvature Surfaces in SXR and HXR. Michigan Math. J. 63, 701-723 (2014).
• [30] Molnár, E.: The projective interpretation of the eight 3-dimensional homogeneous geometries. Beitr. Algebra Geom. 38 No. 2, 261-288 (1977).
• [31] : Molnár, E., Szirmai, J.: Symmetries in the 8 homogeneous 3-geometries. Symmetry Cult. Sci. 21/1-3, 87-117 (2010).
• [32] Molnár, E., Szilágyi, B.: Translation curves and their spheres in homogeneous geometries. Publ. Math. Debrecen. 78/2, 327-346 (2010).
• [33] Molnár, E.: On projective models of Thurston geometries, some relevant notes on Nil orbifolds and manifolds. Sib. Electron. Math. Izv. 7, 491-498 (2010), http://mi.mathnet.ru/semr267.
• [34] Molnár, E., Szirmai, J.: On Nil crystallography. Symmetry Cult. Sci. 17/1-2, 55-74 (2006).
• [35] Molnár, E., Szirmai, J.: Top dense hyperbolic ball packings and coverings for complete Coxeter orthoscheme groups. Publications de l’Institut Mathématique. 103(117), 129-146 (2018), https://doi.org/10.2298/PIM1817129M.
• [36] Molnár, E., Szirmai, J.: Classification of Sol lattices. Geom. Dedicata. 161/1, 251-275 (2012).
• [37] Molnár, E., Szirmai, J., Vesnin, A.: Projective metric realizations of cone-manifolds with singularities along 2-bridge knots and links. J. Geom. 95, 91-133 (2009).
• [38] Molnár, E., Szirmai, J.: Volumes and geodesic ball packings to the regular prism tilings in^SL2R space. Publ. Math. Debrecen. 84(1-2), 189-203 (2014).
• [39] Molnár, E., Szirmai, J., Vesnin, A.: Packings by translation balls in^SL2R. J. Geom. 105(2), 287-306 (2014).
• [40] Molnár E., Szirmai J., Vesnin A.: Geodesic and Translation Ball Packings Generated by Prismatic Tesselations of the Universal Cover of ^SL2R. Results in Math. 71), 623-642 (2017).
• [41] Morabito, F., Rodriguez, M. M.: Classification of rotational special Weingarten surfaces of minimal type in S2×R and H2×R. Mathematische Zeitschrift. 273 (1-2), 379-399 (2013), https://doi.org/10.1007/s00209-012-1010-3.
• [42] Németh, L.: Pascal pyramid in the space H2×R. Mathematical Communications. 22, 211-225 (2017). [43] Novello, T., da Silva, V., Velhoa, L.: Visualization of Nil, Sol, and^SL2R geometries. Computers and Graphics. 91, 219-231 (2020).
• [44] Ohshika K., Papadopoulos, A. (editors): In the Tradition of Thurston Geometry and Topology, Springer International Publishing. (2020), ISBN:978-3-030-55927-4.
• [45] Pallagi, J., Schultz, B., Szirmai, J.: Visualization of geodesic curves, spheres and equidistant surfaces in S2×R space. KoG. 14, 35-40 (2010).
• [46] Pallagi, J., Schultz, B., Szirmai, J.: Equidistant surfaces in Nil space. Stud. Univ. Zilina, Math. Ser. 25, 31-40 (2011).
• [47] Pallagi, J., Schultz, B., Szirmai, J.: Equidistant surfaces in H2×R space. KoG. 15, 3-6 (2011).
• [48] Pallagi, J., Szirmai, J.: Visualization of the Dirichlet-Voronoi cells in S2×R space. Pollack Periodica. 7 Supp 1, 95–104 (2012), https://doi.org/10.1556/Pollack.7.2012.S.9.
• [49] Papadopoulos, A., Su, W.: On hyperbolic analogues of some classical theorems in spherical geometry. hal-01064449 (2014).
• [50] Rodriguez, M. M.: Minimal surfaces with limit ends in H2×R. Journal für die reine und angewandte Mathematik (Crelle’s Journal). 685, 123-141 (2013), https://doi.org/10.1515/crelle-2012-0010.
• [51] Schultz B., Molnár E.: Geodesic lines and spheres, densest(?) geodesic ball packing in the new linear model of Nil geometry. Proceedings of the Czech-Slovak Conference on Geometry and Graphics. 177-186 (2015), ISBN 978-80-227-4479-9.
• [52] Schultz, B., Szirmai, J.: On parallelohedra of Nil-space. Pollack Periodica. 7. Supp 1, 129-136 (2012).
• [53] Schultz, B., Szirmai, J.: Geodesic ball packings generated by regular prism tilings in Nil geometry Miskolc Math. Notes. 23/1, 429-442 (2012), https://doi.org/10.18514/MMN.2022.2959.
• [54] Scott, P.: The geometries of 3-manifolds. Bull. London Math. Soc. 15, 401-487 (1983).
• [55] Szirmai, J.: The densest geodesic ball packing by a type of Nil lattices. Beitr. Algebra Geom. 48(2), 383-398 (2007).
• [56] Szirmai, J.: A candidate to the densest packing with equal balls in the Thurston geometries. Beitr. Algebra Geom. 55(2), 441-452 (2014).
• [57] Szirmai, J.: Lattice-like translation ball packings in Nil space. Publ. Math. Debrecen. 80(3-4), 427-440 (2012).
• [58] Szirmai, J.: Nil geodesic triangles and their interior angle sums. Bull. Braz. Math. Soc. (N.S.) 49, 761-773 (2018).
• [59] Szirmai, J.: Non-periodic geodesic ball packings to infinite regular prism tilings in SL(2,R) space. Rocky Mountain Journal of Mathematics. 46/3, 1055-1070 (2016).
• [60] Szirmai, J.: Bisector surfaces and circumscribed spheres of tetrahedra derived by translation curves in Sol geometry. New York J. Math. 25, 107-122 (2019).
• [61] Szirmai, J.: Simply transitive geodesic ball packings to S2 × R space groups generated by glide reflections. Ann. Mat. Pur. Appl., 193/4, 1201-1211 (2014), https://doi.org/10.1007/s10231-013-0324-z.
• [62] Szirmai, J.: Geodesic ball packings in S2×R space for generalized Coxeter space groups. Beitr. Algebra Geom. 52, 413 - 430 (2011).
• [63] Szirmai, J.: Geodesic ball packings in H2×R space for generalized Coxeter space groups. Math. Commun. 17/1, 151-170 (2012).
• [64] Szirmai, J:. The densest translation ball packing by fundamental lattices in Sol space. Beitr. Algebra Geom. 51(2), 353-373 (2010).
• [65] Szirmai, J.: Interior angle sums of geodesic triangles in S2×R and H2×R geometries. Bul. Acad. de Stiinte Republicii Mold. Mat. 93(2), 44-61 (2020).
• [66] Szirmai, J.: Apollonius surfaces, circumscribed spheres of tetrahedra, Menelaus’ and Ceva’s theorems in S2×R and H2×R geometries. Q. J. Math. 73, 477-494 (2022), https://doi.org/10.1093/qmath/haab038.
• [67] Szirmai, J.: On Menelaus’ and Ceva’s theorem in Nil geometry. Acta Univ. Sapientiae Math. (2023), arXiv: 2110.08877.
• [68] Szirmai, J.: Regular prism tilings in^SL2R space Aequat. Math. 88 (1-2), 67-79 (2014), https://doi.org/10.1007/s00010-013-0221-y
• [69] Szirmai, J.: Hyperball packings in hyperbolic 3-space. Mat. Vesn., 70/3, 211-221 (2018).
• [70] Szirmai, J.: Packings with horo- and hyperballs generated by simple frustum orthoschemes. Acta Math. Hungar. 152/2, 365-382 (2017), https://doi.org/10.1007/s10474-017-0728-0.
• [71] Szirmai, J.: Density upper bound of congruent and non-congruent hyperball packings generated by truncated regular simplex tilings. Rendiconti del Circolo Matematico di Palermo Series 2, 67, 307-322 (2018), https://doi.org/10.1007/s12215-017-0316-8.
• [72] Szirmai, J.: Decomposition method related to saturated hyperball packings. Ars Math. Contemp., 16, 349-358 (2019).
• [73] Szirmai, J.: The optimal ball and horoball packings to the Coxeter honeycombs in the hyperbolic d-space. Beitr. Algebra Geom. 48/1, 35-47 (2007).
• [74] Szirmai, J.: Horoball packings to the totally asymptotic regular simplex in the hyperbolic n-space. Aequat. Math. 85, 471-482 (2013), https://doi.org/10.1007/s00010-012-0158-6.
• [75] Szirmai, J.: Horoball packings and their densities by generalized simplicial density function in the hyperbolic space. Acta Math. Hungar. 136/1-2, 39-55 (2012), https://doi.org/10.1007/s10474-012-0205-8.
• [76] Szirmai, J.: The p-gonal prism tilings and their optimal hypersphere packings in the hyperbolic 3-space. Acta Math. Hungar. 111 (1-2), 65-76 (2006).
• [77] Szirmai, J.: The regular prism tilings and their optimal hyperball packings in the hyperbolic n-space. Publ. Math. Debrecen. 69 (1-2), 195-207 (2006).
• [78] Szirmai, J.: The optimal hyperball packings related to the smallest compact arithmetic 5-orbifolds. Kragujevac J. Math. 40(2), 260-270 (2016), https://doi.org/10.5937/KgJMath1602260S.
• [79] Szirmai, J.: The least dense hyperball covering to the regular prism tilings in the hyperbolic n-space. Ann. Mat. Pur. Appl. 195/1, 235-248 (2016), https://doi.org/10.1007/s10231-014-0460-0.
• [80] Thurston,W. P. (and Levy, S. editor): Three-Dimensional Geometry and Topology. Princeton University Press, Princeton, New Jersey, vol. 1 (1997).
• [81] Vránics, A., Szirmai, J.: Lattice coverings by congruent translation balls using translation-like bisector surfaces in Nil Geometry. KoG. 23, 6-17 (2019).
• [82] Weeks, J. R.: Real-time animation in hyperbolic, spherical, and product geometries. A. Prékopa and E. Molnár, (eds.). Non-Euclidean Geometries, János Bolyai Memorial Volume, Mathematics and Its Applications, Springer Vol. 581, 287-305 (2006).