Singularities of Gauss maps of wave fronts with non-degenerate singular points

• Autorzy: Teramoto Keisuke

Identyfikatory

DOI

10.4064/ba200820-13-11

• Języki publikacji: EN

Abstrakty

EN

We study singularities of Gauss maps of (wave) fronts and give characterizations of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate the relation between a kind of boundedness of Gaussian curvatures near cuspidal edges and types of singularities of their Gauss maps. Further, we consider extended height functions on fronts with non-degenerate singular points.

Słowa kluczowe

EN

Gauss map; wave front; singularity; extended height function

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2021
• Tom: Vol. 69, no. 2
• Strony: 149--169
• Opis fizyczny: Bibliogr. 43 poz.

Twórcy

autor

Teramoto Keisuke

• Department of Mathematics Hiroshima University Higashi-Hiroshima Hiroshima 739-8526, Japan

Bibliografia

• [1] V. I. Arnol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps. Vol. I. The Classification of Critical Points, Caustics and Wave Fronts, Monogr. Math. 82, Birkhäuser Boston, Boston, MA, 1985.
• [2] T. Banchoff, T. Gaffney and C. McCrory, Cusps of Gauss Mappings, Res. Notes in Math. 55, Pitman, 1981.
• [3] D. Bleecker and L. Wilson, Stability of Gauss maps, Illinois J. Math. 22 (1978), 279-289.
• [4] J. W. Bruce and P. J. Giblin, Curves and Singularities, 2nd ed., Cambridge Univ. Press, 1992.
• [5] J. W. Bruce, P. J. Giblin and F. Tari, Families of surfaces: height functions, Gauss maps and duals, in: Real and Complex Singularities (São Carlos, 1994), Pitman Res. Notes Math. Ser. 333, Longman, Harlow, 1995, 148-178.
• [6] J. W. Bruce, P. J. Giblin and F. Tari, Families of surfaces: height functions and projections to planes, Math. Scand. 82 (1998), 165-185.
• [7] J. W. Bruce, P. J. Giblin and F. Tari, Families of surfaces: focal sets, ridges and umbilics, Math. Proc. Cambridge Philos. Soc. 125 (1999), 243-268.
• [8] J. W. Bruce and F. Tari, Extrema of principal curvature and symmetry, Proc. Edin burgh Math. Soc. 39 (1996), 397-402.
• [9] A. P. Francisco, Functions on a swallowtail, arXiv:1804.09664 (2018).
• [10] S. Fujimori, K. Saji, M. Umehara and K. Yamada, Singularities of maximal surfaces, Math. Z. 259 (2008), 827-848.
• [11] T. Fukui and M. Hasegawa, Singularities of parallel surfaces, Tohoku Math. J. 64 (2012), 387-408.
• [12] T. Fukui and M. Hasegawa, Fronts of Whitney umbrella—a differential geometric approach via blowing up, J. Singularities 4 (2012), 35-67.
• [13] T. Fukui and M. Hasegawa, Height functions on Whitney umbrellas, RIMS Kôkyûroku Bessatsu B38 (2013), 153-168.
• [14] M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities, Grad. Texts in Math. 14, Springer, 1973.
• [15] M. Hasegawa, A. Honda, K. Naokawa, K. Saji, M. Umehara and K. Yamada, Intrinsic properties of surfaces with singularities, Int. J. Math. 26 (2015), art. 1540008, 34 pp.
• [16] A. Honda, K. Naokawa, M. Umehara and K. Yamada, Isometric deformations of wave fronts at non-degenerate singular points, Hiroshima Math. J. 50 (2020), 269-312.
• [17] G. Isihkawa, Singularities of frontals, in: Singularities in Generic Geometry, Adv. Stud. Pure Math. 78, Math. Soc. Japan, 2018, 55-106.
• [18] S. Izumiya, M. C. Romero Fuster, M. A. S. Ruas and F. Tari, Differential Geometry from a Singularity Theory Viewpoint, World Sci., Hackensack, NJ, 2016.
• [19] S. Izumiya and K. Saji, The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and “flat” spacelike surfaces, J. Singularities 2 (2010), 92-127.
• [20] S. Izumiya, K. Saji and M. Takahashi, Horospherical flat surfaces in hyperbolic 3-space, J. Math. Soc. Japan 62 (2010), 789-849.
• [21] S. Izumiya, K. Saji and N. Takeuchi, Flat surfaces along cuspidal edges, J. Singularities 16 (2017), 73-100.
• [22] Y. Kabata, Recognition of plane-to-plane map-germs, Topology Appl. 202 (2016), 216-238.
• [23] M. Kokubu, W. Rossman, K. Saji, M. Umehara and K. Yamada, Singularities of flat fronts in hyperbolic space, Pacific J. Math. 221 (2005), 303-351.
• [24] M. Kokubu and M. Umehara, Orientability of linear Weingarten surfaces, spacelike CMC-1 surfaces and maximal surfaces, Math. Nachr. 284 (2011), 1903-1918.
• [25] L. F. Martins and J. J. Nuño-Ballesteros, Contact properties of surfaces in R3 with corank 1 singularities, Tohoku Math. J. 67 (2015), 105-124.
• [26] L. F. Martins and K. Saji, Geometric invariants of cuspidal edges, Canad. J. Math. 68 (2016), 445-462.
• [27] 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, in: Geometry and Topology of Manifolds, Springer Proc. Math. Statist. 154, Springer, Tokyo, 2016, 247-281.
• [28] S. Murata and M. Umehara, Flat surfaces with singularities in Euclidean 3-space, J. Differential Geom. 221 (2005), 303-351.
• [29] K. Naokawa, M. Umehara and K. Yamada, Isometric deformations of cuspidal edges, Tohoku Math. J. 68 (2016), 73-90.
• [30] R. Oset Sinha and F. Tari, On the flat geometry of the cuspidal edge, Osaka J. Math. 55 (2018), 393-421.
• [31] I. R. Porteous, The normal singularities of a submanifold, J. Differential Geom. 5 (1971), 543-564.
• [32] I. R. Porteous, Geometric Differentiation, Cambridge Univ. Press, 2001.
• [33] J. H. Rieger, Families of maps from the plane to the plane, J. London Math. Soc. (2) 36 (1987), 351-369.
• [34] K. Saji, Criteria for singularities of smooth maps from the plane into the plane and their applications, Hiroshima Math. J. 40 (2010), 229-239.
• [35] K. Saji, Criteria for D4 singularities of wave fronts, Tohoku Math. J. 63 (2011), 137-147.
• [36] K. Saji and K. Teramoto, Dualities of geometric invariants on cuspidal edges on flat fronts in the hyperbolic space and the de Sitter space, Mediterr. J. Math. 17 (2020), art. 42, 20 pp.
• [37] K. Saji, M. Umehara and K. Yamada, Ak singularities of wave fronts, Math. Proc. Cambridge Philos. Soc. 146 (2009), 731-746.
• [38] K. Saji, M. Umehara and K. Yamada, The geometry of fronts, Ann. of Math. 169 (2009), 491-529.
• [39] K. Saji, M. Umehara and K. Yamada, The duality between singular points and inflection points on wave fronts, Osaka J. Math. 47 (2010), 591-607.
• [40] K. Teramoto, Parallel and dual surfaces of cuspidal edges, Differential Geom. Appl. 44 (2016), 52-62.
• [41] K. Teramoto, Focal surfaces of wave fronts in the Euclidean 3-space, Glasgow Math. J. 61 (2019), 425-440.
• [42] K. Teramoto, Principal curvatures and parallel surfaces of wave fronts, Adv. Geom. 19 (2019), 541-554.
• [43] H. Whitney, On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane, Ann. of Math. 62 (1955), 374-410.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).