Caustics of pseudo-spherical surfaces in the Euclidean 3-space

References

[1] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps. Volume 1. Classification of Critical Points, Caustics and Wave Fronts, Mod. Birkhäuser Class., Birkhäuser/Springer, New York, 2012. 10.1007/978-0-8176-8340-5Search in Google Scholar

[2] D. Brander, Pseudospherical surfaces with singularities, Ann. Mat. Pura Appl. (4) 196 (2017), no. 3, 905–928. 10.1007/s10231-016-0601-8Search in Google Scholar

[3] D. Brander and F. Tari, Wave maps and constant curvature surfaces: singularities and bifurcations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), no. 1, 361–397. 10.2422/2036-2145.202002_008Search in Google Scholar

[4] J. W. Bruce and T. C. Wilkinson, Folding maps and focal sets, Singularity Theory and its Applications, Part I (Coventry 1988/1989), Lecture Notes in Math. 1462, Springer, Berlin (1991), 63–72. 10.1007/BFb0086374Search in Google Scholar

[5] S. S. Chern and C. L. Terng, An analogue of Bäcklund’s theorem in affine geometry, Rocky Mountain J. Math. 10 (1980), no. 1, 105–124. 10.1216/RMJ-1980-10-1-105Search in Google Scholar

[6] L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York, 1960. Search in Google Scholar

[7] S. Fujimori, K. Saji, M. Umehara and K. Yamada, Singularities of maximal surfaces, Math. Z. 259 (2008), no. 4, 827–848. 10.1007/s00209-007-0250-0Search in Google Scholar

[8] T. Fukunaga and M. Takahashi, Framed surfaces in the Euclidean space, Bull. Braz. Math. Soc. (N. S.) 50 (2019), no. 1, 37–65. 10.1007/s00574-018-0090-zSearch in Google Scholar

[9] C. Goulart and K. Tenenblat, On Bäcklund and Ribaucour transformations for surfaces with constant negative curvature, Geom. Dedicata 181 (2016), 83–102. 10.1007/s10711-015-0113-5Search in Google Scholar

[10] D. Hilbert, Ueber Flächen von constanter Gaussscher Krümmung, Trans. Amer. Math. Soc. 2 (1901), no. 1, 87–99. 10.1090/S0002-9947-1901-1500557-5Search in Google Scholar

[11] H. Hopf, Differential Geometry in the Large, 2nd ed., Lecture Notes in Math. 1000, Springer, Berlin, 1989. 10.1007/3-540-39482-6Search in Google Scholar

[12] G.-O. Ishikawa and Y. Machida, Singularities of improper affine spheres and surfaces of constant Gaussian curvature, Internat. J. Math. 17 (2006), no. 3, 269–293. 10.1142/S0129167X06003485Search in Google Scholar

[13] S. Izumiya, M. D. C. Romero Fuster, M. A. S. Ruas and F. Tari, Differential Geometry from a Singularity Theory Viewpoint, World Scientific, Hackensack, 2016. Search in Google Scholar

[14] S. Izumiya and K. Saji, The mandala of Legendrian dualities for pseudo-spheres in Lorentz–Minkowski space and “flat” spacelike surfaces, J. Singul. 2 (2010), 92–127. 10.5427/jsing.2010.2gSearch in Google Scholar

[15] S. Izumiya, K. Saji and M. Takahashi, Horospherical flat surfaces in hyperbolic 3-space, J. Math. Soc. Japan 62 (2010), no. 3, 789–849. 10.2969/jmsj/06230789Search in Google Scholar

[16] S. Izumiya, K. Saji and N. Takeuchi, Singularities of line congruences, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 6, 1341–1359. 10.1017/S0308210500002973Search in Google Scholar

[17] M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyperbolic space, Pacific J. Math. 221 (2005), no. 2, 303–351. 10.2140/pjm.2005.221.303Search in Google Scholar

[18] L. F. Martins, K. Saji, M. Umehara and K. Yamada, Behavior of Gaussian curvature and mean curvature near non-degenerate singular points on wave fronts, Geometry and Topology of Manifolds, Springer Proc. Math. Stat. 154, Springer, Tokyo (2016), 247–281. 10.1007/978-4-431-56021-0_14Search in Google Scholar

[19] M. Melko and I. Sterling, Application of soliton theory to the construction of pseudospherical surfaces in 𝐑 3 , Ann. Global Anal. Geom. 11 (1993), no. 1, 65–107. 10.1007/BF00773365Search in Google Scholar

[20] R. Morris, The sub-parabolic lines of a surface, The mathematics of Surfaces, VI (Uxbridge 1994), Inst. Math. Appl. Conf. Ser. New Ser. 58, Oxford University, New York (1996), 79–102. Search in Google Scholar

[21] S. Murata and M. Umehara, Flat surfaces with singularities in Euclidean 3-space, J. Differential Geom. 82 (2009), no. 2, 279–316. 10.4310/jdg/1246888486Search in Google Scholar

[22] R. S. Palais and C.-L. Terng, Critical Point Theory and Submanifold Geometry, Lecture Notes in Math. 1353, Springer, Berlin, 1988. 10.1007/BFb0087442Search in Google Scholar

[23] K. Saji, Criteria for cuspidal S k singularities and their applications, J. Gökova Geom. Topol. GGT 4 (2010), 67–81. Search in Google Scholar

[24] K. Saji, Criteria for D 4 singularities of wave fronts, Tohoku Math. J. (2) 63 (2011), no. 1, 137–147. 10.2748/tmj/1303219939Search in Google Scholar

[25] K. Saji and K. Teramoto, Behavior of principal curvatures of frontals near non-front singular points and their applications, J. Geom. 112 (2021), no. 3, Paper No. 39. 10.1007/s00022-021-00605-3Search in Google Scholar

[26] K. Saji, M. Umehara and K. Yamada, A k singularities of wave fronts, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 3, 731–746. 10.1017/S0305004108001977Search in Google Scholar

[27] K. Saji, M. Umehara and K. Yamada, The geometry of fronts, Ann. of Math. (2) 169 (2009), no. 2, 491–529. 10.4007/annals.2009.169.491Search in Google Scholar

[28] K. Saji, M. Umehara and K. Yamada, A 2 -singularities of hypersurfaces with non-negative sectional curvature in Euclidean space, Kodai Math. J. 34 (2011), no. 3, 390–409. 10.2996/kmj/1320935549Search in Google Scholar

[29] K. Saji, M. Umehara and K. Yamada, Coherent tangent bundles and Gauss–Bonnet formulas for wave fronts, J. Geom. Anal. 22 (2012), no. 2, 383–409. 10.1007/s12220-010-9193-5Search in Google Scholar

[30] R. Sasaki, Soliton equations and pseudospherical surfaces, Nuclear Phys. B 154 (1979), no. 2, 343–357. 10.1016/0550-3213(79)90517-0Search in Google Scholar

[31] M. D. Shepherd, Line congruences as surfaces in the space of lines, Differential Geom. Appl. 10 (1999), no. 1, 1–26. 10.1016/S0926-2245(98)00025-4Search in Google Scholar

[32] M. Takahashi and K. Teramoto, Surfaces of revolution of frontals in the Euclidean space, Bull. Braz. Math. Soc. (N. S.) 51 (2020), no. 4, 887–914. 10.1007/s00574-019-00180-xSearch in Google Scholar

[33] T. A. M. Tejeda, Extendibility and boundedness of invariants on singularities of wavefronts, preprint (2020), https://arxiv.org/abs/2011.09511. Search in Google Scholar

[34] K. Teramoto, Focal surfaces of wave fronts in the Euclidean 3-space, Glasg. Math. J. 61 (2019), no. 2, 425–440. 10.1017/S0017089518000277Search in Google Scholar

[35] K. Teramoto, Principal curvatures and parallel surfaces of wave fronts, Adv. Geom. 19 (2019), no. 4, 541–554. 10.1515/advgeom-2018-0038Search in Google Scholar

[36] K. Teramoto, Focal surfaces of fronts associated to unbounded principal curvatures, preprint (2022), https://arxiv.org/abs/2210.06221; to appear in Rocky Mountain J. Math. 10.1216/rmj.2023.53.1587Search in Google Scholar

[37] C.-L. Terng and K. Uhlenbeck, Geometry of solitons, Notices Amer. Math. Soc. 47 (2000), no. 1, 17–25. Search in Google Scholar

[38] T. Ura, Constant negative Gaussian curvature tori and their singularities, Tsukuba J. Math. 42 (2018), no. 1, 65–95. 10.21099/tkbjm/1541559651Search in Google Scholar

[39] R. C. Yates, A Handbook on Curves and Their Properties, J. W. Edwards, Ann Arbor, 1947. Search in Google Scholar