Caustics of pseudo-spherical surfaces in the Euclidean 3-space. (English) Zbl 1536.53018
The author studies geometric properties of caustics of pseudo-spherical surfaces, that is, surfaces with constant negative Gaussian curvature \(-1\) in the Euclidean \(3\)-space \(\mathbb R^3\). First, he recalls some properties of fronts and their singularities, the notion of sine-Gordon equation which determines pseudo-spherical surfaces, discusses the notion of moving frames and frame equation, singularities of pseudo-spherical fronts and singularities of caustics of such fronts, Gaussian curvature and mean curvature of the caustics and so on. Among other things, he characterizes singularities of pseudo-spherical fronts and their caustics and also obtains explicit descriptions for the Gaussian and mean curvature in terms of the solution of the sine-Gordon equation under certain conditions. Above all, the author formulates a specific condition under which caustics of pseudo-spherical fronts become minimal surfaces, and then identifies a class of such pseudo-spherical fronts. In conclusion, he demonstrates an isometric deformation from the catenoid to the helicoid via a family of caustics of Dini surfaces with certain parametrizations using a series of nice pictures (see also [C.-L. Terng and K. Uhlenbeck, Notices Am. Math. Soc. 47, No. 1, 17–25 (2000; Zbl 0987.37072); M. Takahashi and the author, Bull. Braz. Math. Soc. (N.S.) 51, No. 4, 887–914 (2020; Zbl 1475.57041)]).
Reviewer: Aleksandr G. Aleksandrov (Moskva)
MSC:
| 53A05 | Surfaces in Euclidean and related spaces |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 58J72 | Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds |
| 57R45 | Singularities of differentiable mappings in differential topology |
Keywords:
pseudo-spherical surfaces; caustics; fronts; minimal surfaces; sine-Gordon equation; Bäcklund transformation; tractrix; catenoid; helicoidSoftware:
SingSurfReferences:
| [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. · Zbl 1290.58001 |
| [2] | D. Brander, Pseudospherical surfaces with singularities, Ann. Mat. Pura Appl. (4) 196 (2017), no. 3, 905-928. · Zbl 1371.53003 |
| [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. · Zbl 1503.53034 |
| [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. · Zbl 0733.58002 |
| [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. · Zbl 0407.53002 |
| [6] | L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, Dover, New York, 1960. · JFM 40.0626.02 |
| [7] | S. Fujimori, K. Saji, M. Umehara and K. Yamada, Singularities of maximal surfaces, Math. Z. 259 (2008), no. 4, 827-848. · Zbl 1145.57026 |
| [8] | T. Fukunaga and M. Takahashi, Framed surfaces in the Euclidean space, Bull. Braz. Math. Soc. (N. S.) 50 (2019), no. 1, 37-65. · Zbl 1425.58021 |
| [9] | C. Goulart and K. Tenenblat, On Bäcklund and Ribaucour transformations for surfaces with constant negative curvature, Geom. Dedicata 181 (2016), 83-102. · Zbl 1360.53013 |
| [10] | D. Hilbert, Ueber Flächen von constanter Gaussscher Krümmung, Trans. Amer. Math. Soc. 2 (1901), no. 1, 87-99. · JFM 32.0608.01 |
| [11] | H. Hopf, Differential Geometry in the Large, 2nd ed., Lecture Notes in Math. 1000, Springer, Berlin, 1989. · Zbl 0669.53001 |
| [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. · Zbl 1093.53067 |
| [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. · Zbl 1369.53004 |
| [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. · Zbl 1292.53009 |
| [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. · Zbl 1205.53065 |
| [16] | S. Izumiya, K. Saji and N. Takeuchi, Singularities of line congruences, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 6, 1341-1359. · Zbl 1053.53007 |
| [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. · Zbl 1110.53044 |
| [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. · Zbl 1347.53044 |
| [19] | M. Melko and I. Sterling, Application of soliton theory to the construction of pseudospherical surfaces in \(\mathbf{R}}^{3\), Ann. Global Anal. Geom. 11 (1993), no. 1, 65-107. · Zbl 0810.53003 |
| [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. · Zbl 0960.53006 |
| [21] | S. Murata and M. Umehara, Flat surfaces with singularities in Euclidean 3-space, J. Differential Geom. 82 (2009), no. 2, 279-316. · Zbl 1184.53015 |
| [22] | R. S. Palais and C.-L. Terng, Critical Point Theory and Submanifold Geometry, Lecture Notes in Math. 1353, Springer, Berlin, 1988. · Zbl 0658.49001 |
| [23] | K. Saji, Criteria for cuspidal S_k singularities and their applications, J. Gökova Geom. Topol. GGT 4 (2010), 67-81. · Zbl 1267.58023 |
| [24] | K. Saji, Criteria for D_4 singularities of wave fronts, Tohoku Math. J. (2) 63 (2011), no. 1, 137-147. · Zbl 1233.57017 |
| [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. · Zbl 1487.57041 |
| [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. · Zbl 1173.53039 |
| [27] | K. Saji, M. Umehara and K. Yamada, The geometry of fronts, Ann. of Math. (2) 169 (2009), no. 2, 491-529. · Zbl 1177.53014 |
| [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. · Zbl 1272.57022 |
| [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. · Zbl 1276.57032 |
| [30] | R. Sasaki, Soliton equations and pseudospherical surfaces, Nuclear Phys. B 154 (1979), no. 2, 343-357. |
| [31] | M. D. Shepherd, Line congruences as surfaces in the space of lines, Differential Geom. Appl. 10 (1999), no. 1, 1-26. · Zbl 0953.53012 |
| [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. · Zbl 1475.57041 |
| [33] | T. A. M. Tejeda, Extendibility and boundedness of invariants on singularities of wavefronts, preprint (2020), https://arxiv.org/abs/2011.09511. |
| [34] | K. Teramoto, Focal surfaces of wave fronts in the Euclidean 3-space, Glasg. Math. J. 61 (2019), no. 2, 425-440. · Zbl 1429.57028 |
| [35] | K. Teramoto, Principal curvatures and parallel surfaces of wave fronts, Adv. Geom. 19 (2019), no. 4, 541-554. · Zbl 1439.57051 |
| [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. |
| [37] | C.-L. Terng and K. Uhlenbeck, Geometry of solitons, Notices Amer. Math. Soc. 47 (2000), no. 1, 17-25. · Zbl 0987.37072 |
| [38] | T. Ura, Constant negative Gaussian curvature tori and their singularities, Tsukuba J. Math. 42 (2018), no. 1, 65-95. · Zbl 1498.53006 |
| [39] | R. C. Yates, A Handbook on Curves and Their Properties, J. W. Edwards, Ann Arbor, 1947. |
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