Abstract
The simplest way to have birth of surfaces is through transitions in the fibres of a function f with a Morse singularity of index 0 or 3. It is natural to seek to understand the geometry of newly born surfaces. We consider here the question of finding how many umbilics are on a newly born surface. We show that newly born surfaces in the Euclidean 3-space have exactly 4 umbilic points all of type lemon, provided that the Hessian of f at the singular point has pairwise distinct eigenvalues. This is true in both cases when f is an analytic or a smooth germ. When only two of such eigenvalues are equal, the number of umbilic points is either 2, 4, 6 or 8 when f is an analytic or a generic smooth germ. The same results holds for newly born surfaces in the Minkowski 3-space. In that case when the two eigenvalues associated to the two spacelike eigenvectors are distinct we get exactly 4 umbilic points all of type lemon. If they are equal, the number of umbilic points is either 2, 4, 6 or 8.
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Arnol’d, V.I., Guseĭn-Zade, S.M., Varchenko, A.N.: Singularities of differentiable maps. Vol. I. In: The Classification of Critical Points, Caustics and Wave Fronts. Monographs in Mathematics, vol. 82. Birkhauser, Boston (1985)
Bröcker, Th.: Differentiable germs and catastrophes. Translated from the German, last chapter and bibliography by L. Lander. In: London Mathematical Society Lecture Note Series, No. 17. Cambridge University Press, Cambridge (1975)
Bruce, J.W., Fidal, D.: On binary differential equations and umbilics. Proc. R. Soc. Edinb. 111A, 147–168 (1989)
Bruce, J.W., Kirk, N.P., du Plessis, A.A.: Complete transversals and the classification of singularities. Nonlinearity 10, 253–275 (1997)
Bruce, J.W., Tari, F.: On binary differential equations. Nonlinearity 82, 255–271 (1995)
Darboux, G.: Sur la forme des lignes de courbure dans la voisinage d’un ombilic, Leçons sur la Theorie des Surfaces, IV, Note 7. Gauthier Villars, Paris (1896)
Duval, D.: Rational Puiseux expansions. Compos. Math. 70, 119–154 (1989)
Garcia, R., Sotomayor, J.: Differential equations of classical geometry, a qualitative theory. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications] 27o Colóquio Brasileiro de Matemática. [27th Brazilian Mathematics Colloquium] IMPA, Rio de Janeiro (2009)
Garcia, R., Tejada, D.: Principal lines on the ellipsoid in Minkowski space \(\mathbb{R}^3_1\) (2016) (Preprint )
Genin, D., Khesin, B., Tabachnikov, S.: Geodesics on an ellipsoid in Minkowski space. Enseign. Math. 53, 307–331 (2007)
Ghomi, M., Howard, R.: Normal curvatures of asymptotically constant graphs and Carathodory’s conjecture. Proc. Am. Math. Soc. 140, 4323–4335 (2012)
Gutierrez, C., Sotomayor, J.: Jorge lines of curvature, umbilic points and Carathéodory conjecture. Resenhas 3, 291–322 (1998)
Hasegawa, M.: Parabolic, ridge and sub-parabolic curves of implicit surfaces with singularities. To appear in Osaka J. Math
Izumiya, S., Tari, F.: Self-adjoint operators on surfaces with a singular metric. J. Dyn. Control Syst. 16, 329–353 (2010)
Pei, D.: Singularities of \(\mathbb{R}P^2\)-valued Gauss maps of surfaces in Minkowski 3-space. Hokkaido Math. J. 28, 97–115 (1999)
Porteous, I.R.: The normal singularities of surfaces in \(\mathbb{R}^3\). In: Singularities, Part 2 (Arcata, Calif., 1981), vol. 40, pp. 379–393, Proceedings of Symposia in Pure Mathematics. American Mathematical Society, Providence (1983)
Rozoy, L.: Résultats nouveaux à propos des conjectures de Lichnerowicz et de Carathéodory. Séminaire de Théorie Spectrale et Géométrie, vol. 9, 1990–1991, pp. 13–61. Univ. Grenoble I (1991)
Sotomayor, J., Gutierrez, C.: Structurally stable configurations of lines of principal curvature. In: Bifurcation, Ergodic Theory and Applications (Dijon, 1981), 195–215, Astérisque, pp. 98–99. Soc. Math, France (1982)
Tari, F.: Umbilics of surfaces in the Minkowski 3-space. J. Math. Soc. Japan 65, 723–731 (2013)
Wahl, J.M.: Derivations, automorphisms and deformations of quasihomogeneous singularities. In: Singularities, Part 2 (Arcata, Calif., 1981), vol. 40, pp. 613–624, Proceedings of Symposia in Pure Mathematics. American Mathematical Society, Providence (1983)
Wall, C.T.C.: Singular Points of Plane Curves. London Mathematical Society Student Texts, vol. 63. Cambridge University Press, Cambridge (2004)
Acknowledgements
The authors are very grateful to the referee for a careful reading of the paper. Part of the work in this paper was carried out while the second author was a visiting professor at Northeastern University. He would like to thank Terry Gaffney and David Massey for their hospitality during his visit.
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The first author was supported by the FAPESP post-doctoral grant number 2013/02543-1 during a post-doctoral period at ICMC-USP, São Carlos, Brazil. The second author was partially supported by the grants FAPESP 2016/02701-4, 2014/00304-2 and CNPq 302956/2015-8, 472796/2013-5.
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Hasegawa, M., Tari, F. On Umbilic Points on Newly Born Surfaces. Bull Braz Math Soc, New Series 48, 679–696 (2017). https://doi.org/10.1007/s00574-017-0037-9
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DOI: https://doi.org/10.1007/s00574-017-0037-9