Inflection points and topology of surfaces in 4-space
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- by Ronaldo Alves Garcia, Dirce Kiyomi Hayashida Mochida, Maria del Carmen Romero Fuster and Maria Aparecida Soares Ruas PDF
- Trans. Amer. Math. Soc. 352 (2000), 3029-3043 Request permission
Abstract:
We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincaré-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.References
- Yannick Louis Kergosien and René Thom, Sur les points paraboliques des surfaces, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 15, A705–A710 (French, with English summary). MR 574308
- J. W. Bruce and F. Tari, On binary differential equations, Nonlinearity 8 (1995), no. 2, 255–271. MR 1328597
- J. W. Bruce and F. Tari, Generic $1$-parameter families of binary differential equations of Morse type, Discrete Contin. Dynam. Systems 3 (1997), no. 1, 79–90. MR 1422540, DOI 10.3934/dcds.1997.3.79
- J. W. Bruce and D. L. Fidal, On binary differential equations and umbilics, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), no. 1-2, 147–168. MR 985996, DOI 10.1017/S0308210500025087
- Lak Dara, Singularités génériques des équations différentielles multiformes, Bol. Soc. Brasil. Mat. 6 (1975), no. 2, 95–128 (French). MR 488153, DOI 10.1007/BF02584779
- G. Darboux, Sur la forme des lignes de courbure dans le voisinage d’un ombilic, Leçons de sur la Théorie des Surfaces, IV, Note 7, Gauthiers-Villars, Paris, 1896.
- A. A. Davydov, The normal form of a differential equation, that is not solved with respect to the derivative, in the neighborhood of its singular point, Funktsional. Anal. i Prilozhen. 19 (1985), no. 2, 1–10, 96 (Russian). MR 800916
- E. A. Feldman, On parabolic and umbilic points of immersed hypersurfaces, Trans. Amer. Math. Soc. 127 (1967), 1–28. MR 206974, DOI 10.1090/S0002-9947-1967-0206974-1
- Víctor Guíñez, Positive quadratic differential forms and foliations with singularities on surfaces, Trans. Amer. Math. Soc. 309 (1988), no. 2, 477–502. MR 961601, DOI 10.1090/S0002-9947-1988-0961601-4
- J. Sotomayor and C. Gutierrez, Structurally stable configurations of lines of principal curvature, Bifurcation, ergodic theory and applications (Dijon, 1981) Astérisque, vol. 98, Soc. Math. France, Paris, 1982, pp. 195–215. MR 724448
- John A. Little, On singularities of submanifolds of higher dimensional Euclidean spaces, Ann. Mat. Pura Appl. (4) 83 (1969), 261–335. MR 271970, DOI 10.1007/BF02411172
- J. Llibre and Y. Yanquian, On the dynamics of surface vector fields and homeomorphisms. Bulletin de la Societé Mathématique de Belgique (to appear).
- Dirce Kiyomi Hayashida Mochida, Maria Del Carmen Romero Fuster, and Maria Aparecida Soares Ruas, The geometry of surfaces in $4$-space from a contact viewpoint, Geom. Dedicata 54 (1995), no. 3, 323–332. MR 1326737, DOI 10.1007/BF01265348
- D. Mond, Thesis, Liverpool (1982).
- R. J. Morris, The use of computer graphics for solving problems in singularity theory. Visualization and Mathematics, Edited by H. Hege and K. Polthie, Springer-Verlag, Heidelberg, 1997, 53–66.
- A. C. Nabarro, Equações Diferenciais Binárias e Geometria Diferencial, M.Sc. thesis, ICMSC-USP, (1997).
- M. C. Romero Fuster, Sphere stratifications and the Gauss map, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), no. 1-2, 115–136. MR 723100, DOI 10.1017/S0308210500015821
- V. D. Sedykh, Some invariants of convex manifolds, Mat. Contemp. 5 (1993), 187–198. Workshop on Real and Complex Singularities (São Carlos, 1992). MR 1305181
- Brian Smyth and Frederico Xavier, A sharp geometric estimate for the index of an umbilic on a smooth surface, Bull. London Math. Soc. 24 (1992), no. 2, 176–180. MR 1148679, DOI 10.1112/blms/24.2.176
Additional Information
- Ronaldo Alves Garcia
- Affiliation: Instituto de Matemática e Estatistica, Universidade Federal de Goiás, 74001-970, Goiânia, GO, Brazil
- Email: ragarcia@mat.ufg.br
- Dirce Kiyomi Hayashida Mochida
- Affiliation: Departamento de Matemática, Universidade Federal de São Carlos, 13560-905, São Carlos, SP, Brazil
- Email: dirce@dm.ufscar.br
- Maria del Carmen Romero Fuster
- Affiliation: Departamento de Geometria e Topologia, Universidad de Valencia, 46000, Valencia, Spain
- MR Author ID: 150115
- Email: romero@uv.es
- Maria Aparecida Soares Ruas
- Affiliation: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Departamento de Matemática, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil
- MR Author ID: 239264
- ORCID: 0000-0001-8890-524X
- Email: maasruas@icmsc.sc.usp.br
- Received by editor(s): September 29, 1997
- Received by editor(s) in revised form: June 15, 1998
- Published electronically: March 15, 2000
- Additional Notes: Research of the first author was partially supported by CNPq and FUNAPE, Brazil.
Research of the third author was partially supported by DGCYT, grant no. PB96-0785
Research of the fourth author was partially supported by CNPq, Brazil, grant # 300066/88-0. - © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 3029-3043
- MSC (1991): Primary 58C27; Secondary 53A05
- DOI: https://doi.org/10.1090/S0002-9947-00-02404-1
- MathSciNet review: 1638242