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A Global View on Generic Geometry

  • María del Carmen Romero Fuster
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 222)

Abstract

We describe how the study of the singularities of height and distance squared functions on submanifolds of Euclidean space, combined with adequate topological and geometrical tools, shows to be useful to obtain global geometrical properties. We illustrate this with several results concerning closed curves and surfaces immersed in \(\mathbb {R}^n\) for \(n=3,4, 5\).

Keywords

Stratifications Height functions Distance squared functions Curvature locus Vertices Semiumbilics Inflection points Convexity 2-regular immersions. 

MS classification

58K05 58C27 53C42 57R30 

References

  1. 1.
    Arnol’d, V.I., Guseĭn-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps. Vol. I. The Classification of Critical Points, Caustics and Wave Fronts. Monographs in Mathematics, vol. 82. Birkhuser (1985)Google Scholar
  2. 2.
    Arnold, V.I.: Singularity Theory. Selected Papers. LMS Lecture Notes Series, vol. 53. Cambridge University Press, Cambridge (1981)Google Scholar
  3. 3.
    Banchoff, T., Gaffney, T., McCrory, C.: Counting tritangent planes of space curves. Topology 24, 15–23 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Banchoff, T., Farris, F.: Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space. Pac. J. Math. 161, 1–24 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Banchoff, T., Gaffney, T., McCrory, C.: Cusps of Gauss Mappings. Research Notes in Mathematics, vol. 55. Pitman (Advanced Publishing Program), Boston (1982)Google Scholar
  6. 6.
    Binotto, R.R., Costa, S.I.R., Romero Fuster, M.C.: The curvature veronese of a 3-manifold immersed in Euclidean space. In: Real and Complex Singularities. Contemporary Maths. vol. 675, pp. 25–44. American Mathematical Society, Providence, RI (2016)Google Scholar
  7. 7.
    Bol, G.: Über Nabelpunkte auf einer Eifläche. Math. Z. 49, 399–410 (1943/44)Google Scholar
  8. 8.
    Bruce, J.W., Giblin, P.J.: Curves and Singularities. Cambridge University Press, Cambridge (1992)Google Scholar
  9. 9.
    Bruce, J.W., Tari, F.: Families of surfaces in \({\mathbb{R}}^4\). Proc. Edinb. Math. Soc. (2) 45(1), 181–203 (2002)Google Scholar
  10. 10.
    Chen, B.Y., Yano, K.: Umbilical submanifolds with respect to a nonparallel normal direction. J. Differ. Geom. 8, 589–597 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Costa, S.I.R.: On closed twisted curves. Proc. Am. Math. Soc. 109, 205–214 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Costa, S.I.R., Romero Fuster, M.C.: Nowhere vanishing torsion closed curves always hide twice. Geom. Dedicata 66, 1–17 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Costa, S.I.R., Moraes, S.M., Romero-Fuster, M.C.: Geometric contacts of surfaces immersed in \({\mathbb{R}}^n, n\ge 5\). Differ. Geom. Appl. 27, 442–454 (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Dreibelbis, D.: Bitangencies on surfaces in four dimensions. Q. J. Math. 52(2), 137–160 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dreibelbis, D.: Singularities of the Gauss map and the binormal surface. Adv. Geom. 3(4), 453–468 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dreibelbis, D.: Invariance of the diagonal contribution in a bitangency formula. In: Real and Complex Singularities. Contemporary Mathematics, vol. 354, pp. 45–56. American Mathematical Society, Providence, RI (2004)Google Scholar
  17. 17.
    Dreibelbis, D.: Birth of bitangencies in a family of surfaces in R4. Differ. Geom. Appl. 24(4), 321–331 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dreibelbis, D.: Self-conjugate vectors of immersed 3-manifolds in \({\mathbb{R}}^6\). Topol. Appl. 159, 450–456 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Feldman, E.A.: The geometry of immersions. I. Trans. Am. Math. Soc. 120, 185–224 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Feldman, E.A.: On parabolic and umbilic points of immersed hypersurfaces. Trans. Am. Math. Soc. 127, 1–28 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Freedman, M.H.: Planes triply tangent to curves with non-vanishing torsion. Topology 19, 1–8 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Garcia, R., Mochida, D.K.H., Romero-Fuster, M.C., Ruas, M.A.S.: Inflection points and topology of surfaces in \(4\)-space. Trans. Am. Math. Soc. 352, 3029–3043 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Garcia, R., Sotomayor, J.: Lines of axial curvature on surfaces immersed in \({{\mathbb{R}}}^4\). Differ. Geom. Appl. 12, 253–269 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Garcia, R., Mello, L.F., Sotomayor, J.: Principal mean curvature foliations on surfaces immersed in \({\mathbb{R}}^4\). EQUADIFF 2003 (2005), pp. 939–950. World Science Publisher, Hackensack, NJGoogle Scholar
  25. 25.
    Ghomi, M.: Tangent lines, inflections and vertices of closed curves. Duke Math. J. 162, 2691–2730 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ghomi, M.: Vertices of closed curves in Riemannian surfaces. Comment. Math. Helv. 88, 427–448 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ghomi, M., Howard, R.: Normal curvatures of asymptotically constant graphs and Carathéodory’s conjecture. Proc. Am. Math. Soc. 140(12), 4323–4335 (2012)CrossRefzbMATHGoogle Scholar
  28. 28.
    Gibson, C.G., Wirthmuller, K., du Plessis, A.A., Looijenga, E.J.N.: Topological Stability of Smooth Mappings. Lecture Notes in Mathematics, vol. 552. Springer, Berlin (1976)Google Scholar
  29. 29.
    Golubitsky, M., Guillemin, V.: Stable Mappings and their Singularities. Springer GTM 14, Berlin (1973)Google Scholar
  30. 30.
    Gonçalves, R.A., Martínez Alfaro, J.A., Montesinos Amilibia, A., Romero-Fuster, M.C.: Relative mean curvature configurations for surfaces in \({\mathbb{R}}^n, n \ge 5\). Bull. Braz. Math. Soc., New Ser. 38(2), 1–22 (2007)Google Scholar
  31. 31.
    Guilfoyle, B., Klingenberg, W.: Proof of the Carathéodory conjecture by mean curvature flow in the space of oriented affine lines. arXiv:0808.0851v1
  32. 32.
    Guíñez, V.: Positive quadratic differential forms and foliations with singularities on surfaces. Trans. Am. Math. Soc. 309(2), 477–502 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Gutierrez, C., Guadalupe, I., Tribuzy, R., Guíñez.: Lines of curvature on surfaces immersed in \({\mathbb{R}}^4\). Bol. Soc. Brasil. Mat. (N.S.) 28, 233–251 (1997)Google Scholar
  34. 34.
    Gutierrez, C., Guadalupe, I., Tribuzy, R., Guíñez, V.: A differential equation for lines of curvature on surfaces immersed in \({{\mathbb{R}}}^4\). Qual. Theory Dyn. Syst. 2, 207–220 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gutierrez, C., Guíñez, V.: Simple umbilic points on surfaces immersed in \({{\mathbb{R}}}^4\). Discret. Contin. Dyn. Syst. 9, 877–900 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Gutiérrez, C., Mercuri, F., Sánchez-Bringas, F.: On a conjecture of Carathéodory: analyticity versus smoothness. Exp. Math. 5(1), 33–37 (1996)CrossRefzbMATHGoogle Scholar
  37. 37.
    Gutiérrez, C., Ruas, M. A. S., Indices of Newton non-degenerate vector fields and a conjecture of Loewner for surfaces in \({\mathbb{R}}^4\). Real and complex singularities. Lecture Notes in Pure and Applied Mathematics, vol. 232, pp. 245–253. Dekker, New York (2003)Google Scholar
  38. 38.
    Gutierrez, C., Sánchez-Bringas, F.: On a Loewner umbilic-index conjecture for surfaces immersed in \( {{\mathbb{R}}}^4\). J. Dyn. Control Syst. 4, 127–136 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Gutiérrez, C., Sánchez-Bringas, F.: Planar vector field versions of Carathéodory’s and Loewner’s conjectures. In: Proceedings of the Symposium on Planar Vector Fields (Lleida, 1996), vol. 41, no. 1, pp. 169–179. Publ. Mat. (1997)Google Scholar
  40. 40.
    Gutierrez, C., Sotomayor, J.: Lines of curvature, umbilic points and Carathéodory conjecture. Resenhas 3, 291–322 (1998)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Hamburger, H.L.: Beweis einer Caratheodoryschen Vermutung. Teil I. (German) Ann. Math. 41(2), 63–86 (1940)Google Scholar
  42. 42.
    Hamburger, H.L.: Beweis einer Caratheodoryschen Vermutung. II. (German). Acta Math. 73, 175–228 (1941)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Hamburger, H.L.: Beweis einer Caratheodoryschen Vermutung. III. (German). Acta Math. 73, 229–332 (1941)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Haupt, O.: Vierscheitelsätze in der ebenen hyperbolischen Geometrie. Geometriae Dedicata 1, 399–414 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Ivanov, V.V.: An analytic conjecture of Carathéodory. (Russian) Sibirsk. Mat. Zh. 43(2), 314–405 (2002), ii; translation in Siberian Math. J. 43(2), 251–322 (2002)Google Scholar
  46. 46.
    Izumiya, S., Romero Fuster, M.C., Ruas, M.A.S., Tari, F.: Differential Geometry from a Singularity Viewpoint. World Scientific, Singapore (2016)Google Scholar
  47. 47.
    Klotz, T.: On Bol’s proof of Carathéodory’s conjecture. Commun. Pure Appl. Math. 12, 277–311 (1959)CrossRefzbMATHGoogle Scholar
  48. 48.
    Kommerell, K.: Riemannsche Flachen in ebenen Raum von vier Dimensionen. Math. Ann. 60, 546–596 (1905)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Little, J.A.: On singularities of submanifolds of higher dimensional Euclidean spaces. Annali Mat. Pura Appl. 4A(83), 261–336 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Looijenga, E.J.N.: Structural stability of smooth families of \(C^{\infty }\) functions, Thesis, University of Amsterdan (1974)Google Scholar
  51. 51.
    Mello, L.F.: Mean directionally curved lines on surfaces immersed in \({{\mathbb{R}}}^4\). Publ. Mat. 47, 415–440 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Mochida, D.K.H.: Geometria generica de subvariedades em codimensao maior que 1. Ph.D thesis, University of São Paulo (1993)Google Scholar
  53. 53.
    Mochida, D.K.H., Romero Fuster, M.C., Ruas, M.A.S.: The geometry of surfaces in \(4\)-space from a contact viewpoint. Geom. Dedicata. 54, 323–332 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Mochida, D.K.H., Romero Fuster, M.C., Ruas, M.A.S.: Osculating hyperplanes and asymptotic directions of codimension two submanifolds of Euclidean spaces. Geom. Dedicata 77, 305–315 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Mochida, D.K.H., Romero Fuster, M.C., Ruas, M.A.S.: Singularities and duality in the flat geometry of submanifolds of Euclidean spaces. Beiträge Algebra Geom. 42, 137–148 (2001)Google Scholar
  56. 56.
    Mochida, D.K.H., Romero Fuster, M.C., Ruas, M.A.S.: Inflection points and nonsingular embeddings of surfaces in \({{\mathbb{R}}}^5\). Rocky Mountain J. Math. 33, 995–1009 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Mond, D.M.Q.: On the classification of germs of maps from \({{{{\mathbb{R}}}}}^2\) to \({{{{\mathbb{R}}}}}^3\). Proc. Lond. Math. Soc. 50, 333–369 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Mond, D.M.Q.: The Classification of Germs of Maps from Surfaces to \(3\)-space, with Applications to the Differential Geometry of Immersions. Ph.D Thesis, Liverpool University (1982)Google Scholar
  59. 59.
    Montaldi, J.A.: Contact with applications to submanifolds. Ph.D Thesis, University of Liverpool (1983)Google Scholar
  60. 60.
    Montaldi, J.A.: On contact between submanifolds. Michigan Math. J. 33, 195–199 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Montaldi, J.A.: On generic composites of maps. Bull. Lond. Math. Soc. 23, 81–85 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Monera, M.G., Montesinos Amilibia, A., Sanabria Codesal, E.: The Taylor expansion of the exponential map and geometric applications. Rev. R. Acad. Cienc. Exactas F. Nat. Ser. A Math. RACSAM 108(2), 881–906 (2014)Google Scholar
  63. 63.
    Moraes, S.M.: Elipses de curvatura no estudo de superficies imersas em \({\mathbb{R}}^n, n \ge 5\). Doctoral Thesis, UNICAMP (Brasil) (2002)Google Scholar
  64. 64.
    Moraes, S.M., Romero-Fuster, M.C.: Semiumbilic and 2-regular immersions of surfaces in Euclidean spaces. Rocky Mt. J. Math. 35, 1327–1345 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Moraes, S.M., Romero-Fuster, M.C., Sanchez-Bringas, F.: Principal configurations and umbilicity of submanifolds in \({{\mathbb{R}}}^N\). Bull. Belg. Math. Soc. Simon Stevin 11(2), 227–245 (2004)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Nabarro, A.C., Romero-Fuster, M.C.: \(3\)-manifolds in Euclidean space from a contact viewpoint. Commun. Anal. Geom. 17, 755–776 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Nikolaev, I.: Foliations on Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A, Series of Modern Surveys in Mathematics. Springer, Berlin (2001)Google Scholar
  68. 68.
    Nuño Ballesteros, J.J.: Submanifolds with a non-degenerate parallel normal vector field in Euclidean spaces. In: Singularity Theory and its Applications. Advanced Studies in Pure Mathematics, vol. 43, pp. 311–332. Mathematical Society of Japan, Tokyo (2006)Google Scholar
  69. 69.
    Nuño Ballesteros, J.J., Romero Fuster, M.C.: A four vertex theorem for strictly convex space curves. J. Geom. 46, 119–126 (1992)Google Scholar
  70. 70.
    Nuño Ballesteros, J.J., Romero Fuster, M.C.: Global bitangency properties of generic closed space curves. Math. Proc. Camb. Philos. Soc. 112(3), 519–526 (1992)Google Scholar
  71. 71.
    Nuño Ballesteros, J.J., Romero-Fuster, M.C.: Contact properties of codimension \(2\) submanifolds with flat normal bundle. Rev. Mat. Iberoam. 26, 799–824 (2010)Google Scholar
  72. 72.
    Nuño Ballesteros, J.J., Romero-Fuster, M.C., Sánchez Bringas, F.: Curvature locus and principal configurations of submanifolds in Euclidean space. To appear in Revista Iberoamericana de MatemáticasGoogle Scholar
  73. 73.
    Osserman, R.: The four-or-more vertex theorem. Am. Math. Mon. 92(5), 332–337 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Ozawa, T.: The numbers of triple tangencies of smooth space curves. Topology 24, 1–13 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  75. 75.
    Porteous, I.R.: The normal singularities of a submanifold. J. Differ. Geom. 5, 543–564 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Porteous, I.R. (1983), The normal singularities of surfaces in \({\mathbb{R}}^3\). Singularities, Part 2 (Arcata, Calif., 1981). In: Proceedings of Symposia in Pure Mathematics, vol. 40, pp. 379–393. American Mathematical Society, Providence, RI (1983)Google Scholar
  77. 77.
    Porteous, I.R.: Ridges and umbilics of surfaces. The mathematics of surfaces, II (Cardiff, 1986). In: Institute of Mathematics and its Applications Conference Series, New Series, vol. 11, pp. 447–458. Oxford University Press, New York (1987)Google Scholar
  78. 78.
    Pohl, W.F.: Differential geometry of higher order. Topology 1, 169–211 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    Pugh, C.: A generalized Poincaré index formula. Topology 7, 217–226 (1968)Google Scholar
  80. 80.
    Ramírez-Galarza, A.I., Sánchez-Bringas, F.: Lines of curvature near umbilical points on surfaces immersed in \({{\mathbb{R}}}^4\). Ann. Global Anal. Geom. 13, 129–140 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Robertson, S.A., Romero Fuster, M.C.: The convex hull of a hypersurface. Proc. Lond. Math. Soc. 50, 370–384 (1985)Google Scholar
  82. 82.
    Romero Fuster, M.C.: The convex hull of an immersion. Ph.D Thesis, University of Southampton (1981)Google Scholar
  83. 83.
    Romero Fuster, M.C.: Sphere stratifications and the Gauss map. Proc. Edinb. Math. Soc. 95A, 115–136 (1983)Google Scholar
  84. 84.
    Romero Fuster, M.C.: Convexly generic curves in \({\mathbb{R}}^3\). Geom. Dedicata 28, 7–29 (1988)Google Scholar
  85. 85.
    Romero Fuster, M.C.: Stereographic projections and geometric singularities. Workshop on real and complex singularities (São Carlos, 1996). Mat. Contemp. 12, 167–182 (1997)Google Scholar
  86. 86.
    Romero Fuster, M.C.: Geometric contacts and 2-regularity of surfaces in Euclidean space. In: Singularity Theory, pp. 307–325. World Scientific Publishing, Hackensack, NJ (2007)Google Scholar
  87. 87.
    Romero Fuster, M.C. Ruas, M.A.S., Tari, F.: Asymptotic curves on surfaces in \({\mathbb{R}}^5\). Commun. Contemp. Math. 10, 309–335 (2008)Google Scholar
  88. 88.
    Romero Fuster, M.C., Sanabria Codesal, E.: On the flat ridges of submanifolds of codimension 2 in \({\mathbb{R}}^n\). Proc. Edinb. Math. Soc. 132A, 975–984 (2002)Google Scholar
  89. 89.
    Romero Fuster, M.C., Sanabria Codesal, E.: Lines of curvature, ridges and conformal invariants of hypersurfaces. Beiträge Algebra Geom. 45, 615–635 (2004)Google Scholar
  90. 90.
    Romero Fuster, M.C., Sanabria Codesal, E.: Conformal invariants interpreted in de Sitter space. Mat. Contemp. 35, 205–220 (2008)Google Scholar
  91. 91.
    Romero Fuster, M.C., Sanabria Codesal, E.: Conformal invariants and spherical contacts of surfaces in \({{\mathbb{R}}}^4\). Rev. Mat. Complut. 26, 215–240 (2013)Google Scholar
  92. 92.
    Romero Fuster, M.C., Sánchez-Bringas F.: Umbilicity of surfaces with orthogonal asymptotic lines in \({{\mathbb{R}}}^4\). Differ. Geom. Appl. 16, 213–224 (2002)Google Scholar
  93. 93.
    Romero Fuster, M.C., Sedykh, V.D.: On the number of singularities, zero curvature points and vertices of a simple convex space curve. J. Geom. 52, 168–172 (1995)Google Scholar
  94. 94.
    Romero Fuster, M.C., Sedykh, V.D.: A lower estimate for the number of zero-torsion points of a space curve. Beitrage Algebra Geom. 38, 183–192 (1997)Google Scholar
  95. 95.
    Scherbel, H.: A new proof of Hamburger’s Index Theorem on umbilical points. Dissertation ETH No. 10281Google Scholar
  96. 96.
    Sedyh, V.D.: Structure of the convex hull of a space curve. (Russian). Trudy Sem. Petrovsk. 6, 239–256 (1981)MathSciNetGoogle Scholar
  97. 97.
    Sedyh, V.D.: Double tangent planes to a space curve. (Russian). Sibirsk. Mat. Zh. 30(1), 209–211 (1989); translation in Siberian Math. J. 30(1), 161–162 (1989)Google Scholar
  98. 98.
    Sedykh, V.D.: The four-vertex theorem of a convex space curve. Funct. Anal. Appl. 26, 28–32 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  99. 99.
    Sedykh, V.D.: A relationship between Lagrange and Legendre singularities in stereographic projection. (Russian). Mat. Sb. 185(12), 123–130 (1994); translation in Russian Acad. Sci. Sb. Math. 83(2), 533–540 (1995)Google Scholar
  100. 100.
    Sedykh, V.D.: On Euler characteristics of manifolds of singularities of wave fronts. (Russian). Funktsional. Anal. i Prilozhen. 46, 92–96 (2012); translation in Funct. Anal. Appl. 46, 77–80 (2012)Google Scholar
  101. 101.
    Smyth, B., Xavier, F.: A sharp geometry estimate for the index of an umbilic point on a smooth surface. Bull. Lond. Math. Soc. 24, 176–180 (1992)CrossRefzbMATHGoogle Scholar
  102. 102.
    Sotomayor, J.: Historical Comments on Monge’s Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in \({\mathbb{R}}^3\) (2004). arXiv:math/0411403v1
  103. 103.
    Sotomayor, J., Gutierrez, C.: Structurally stable configurations of lines of principal curvature. Bifurcation, ergodic theory and applications (Dijon, 1981). In: Astérisque, vol. 98–99, pp. 195–215. Soc. Math, France, Paris (1982)Google Scholar
  104. 104.
    Thom, R.: Sur le cut-locus d’une variété plongée. (French) Collection of articles dedicated to S.S. Chern and D.C. Spencer on their sixtieth birthdays. J. Differ. Geom. 6, 577–586 (1972)CrossRefGoogle Scholar
  105. 105.
    Thobergsson, G., Umehara, M.: A unified approach to the four vertex theorems. II. In: Differential and Symplectic Topology of Knots and Curves. American Mathematical Society Translations: Series 2, pp. 229–252. AMS, Providence, RI (1999)Google Scholar
  106. 106.
    Titus, C.J.: A proof of a conjecture of Carathéodory on umbilic points. Acta Math. 131, 43–77 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  107. 107.
    Umehara, M.: A unified approach to the four vertex theorems I. Differential and symplectic topology of knots and curves. AMS Transl. 190, 185–228 (1999)MathSciNetzbMATHGoogle Scholar
  108. 108.
    Wegner, B.: A cyclographic approach to the vertices of plane curves. J. Geom. 50, 186–201 (1944)MathSciNetCrossRefzbMATHGoogle Scholar
  109. 109.
    Wong, Y.C.: A new curvature theory for surfaces in a Euclidean 4-space. Comment. Math. Helv. 26, 152–170 (1952)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat de ValènciaBurjassot (València)Spain

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