Positive quadratic differential forms and foliations with singularities on surfaces
HTML articles powered by AMS MathViewer
- by Víctor Guíñez PDF
- Trans. Amer. Math. Soc. 309 (1988), 477-502 Request permission
Abstract:
To every positive ${C^r}$-quadratic differential form defined on an oriented two manifold is associated a pair of transversal one-dimensional ${C^r}$-foliations with common singularities. An open set of positive ${C^r}$-quadratic differential forms with structural stable associated foliations is characterized and it is proved that this set is dense in the space of positive ${C^\infty }$-quadratic differential forms with ${C^2}$-topology. Also a realization theorem is established.References
-
C. Gutiérrez, Estabilidade estructural para campos de linhas em variedades $bi$-dimensionais, Doctoral Thesis, IMPA, 1974.
- C. Gutiérrez and J. Sotomayor, An approximation theorem for immersions with stable configurations of lines of principal curvature, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 332–368. MR 730276, DOI 10.1007/BFb0061423
- Philip Hartman and Aurel Wintner, On the singularities in nets of curves defined by differential equations, Amer. J. Math. 75 (1953), 277–297. MR 55007, DOI 10.2307/2372454
- John Hubbard and Howard Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), no. 3-4, 221–274. MR 523212, DOI 10.1007/BF02395062
- A. A. Kadyrov, Singular points of differential equations with nonoriented trajectories on a plane, Differentsial′nye Uravneniya 19 (1983), no. 12, 2039–2048 (Russian). MR 729560
- John N. Mather, Stability of $C^{\infty }$ mappings. I. The division theorem, Ann. of Math. (2) 87 (1968), 89–104. MR 232401, DOI 10.2307/1970595
- Jacob Palis Jr. and Welington de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York-Berlin, 1982. An introduction; Translated from the Portuguese by A. K. Manning. MR 669541, DOI 10.1007/978-1-4612-5703-5
- M. M. Peixoto, Structural stability on two-dimensional manifolds, Topology 1 (1962), 101–120. MR 142859, DOI 10.1016/0040-9383(65)90018-2
- Harold Rosenberg, Labyrinths in the disc and surfaces, Ann. of Math. (2) 117 (1983), no. 1, 1–33. MR 683800, DOI 10.2307/2006969
- Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR 568308
- Jean Martinet, Singularities of smooth functions and maps, London Mathematical Society Lecture Note Series, vol. 58, Cambridge University Press, Cambridge-New York, 1982. Translated from the French by Carl P. Simon. MR 671585
- 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
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 309 (1988), 477-502
- MSC: Primary 57R30; Secondary 49F05, 58A10, 58F99
- DOI: https://doi.org/10.1090/S0002-9947-1988-0961601-4
- MathSciNet review: 961601