Abstract
The smooth structure of the net of principal lines near a closed one (principal cycle) is determined in terms of integral expressions evaluated on the cycle, involving the curvatures and their derivatives.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gutierrez, C.; Sotomayor, J.: Structurally stable configurations of lines of principal curvature. Asterisque 98–99 (1982).
Gutierrez, C.; Sotomayor, J.: An approximation theorem for immersions with stable configurations of lines of principal curvature. In: Geometric Dynamics. Proceedings. Lectures Notes in Math. 1007, Springer, (1983).
Gutierrez, C.; Sotomayor, J.: Closed principal lines and bifurcations. Bol. Soc. Brasil. Mat. 17 (1986).
Gutierrez, C.; Sotomayor, J.: Principal lines on surfaces immersed with constant mean curvature. Trans. Amer. Math. Soc., 239 (1986).
Gutierrez, C.; Sotomayor, J.: Lines of Curvature and Umbilic Points on Surfaces. In: 18th Brasilian Math. Colloquium. IMPA, 1991.
Garcia, R.; Sotomayor, J.: Lines of Curvature Near Hyperbolic Principal Cycles. 3.IInt. Conf. on Dynamical Systems, Chile, 1990. Preprint IMPA, 1991.
Golubitsky, M.; Schaeffer, D.: Singularities and Groups in Bifurcation Theory. Vol. 1. Applied Math. Sciences 51, Springer Verlag, 1985.
Levine, H.; Thom, R.: Singularities of Differentiable Mappings. Lectures Notes in Math. 192, Springer, (1971).
Martinet, J.: Singularités des Functions et Applications Différentiables. Monog. de Mat. Pont. Univ. Catol., Rio de Janeiro, N o 1, 1977. English Translation: Singularities of smooth functions and maps. London Math. Soc. Lecture Note Ser., 58, 1982.
Sotomayor, J.: Closed lines of curvature on Weingarten immersions. Ann. Global Anal. Geom. 5 (1987).
Spivak, M.: A Comprehensive introduction to differential Geometry. Vol. 3. Publish or Perish, Berkeley 1979.
Struick, D.: Lectures on classical differential geometry. Addison Wesley, 1950. Reprinted by Dover, New York, 1988.
Takens, F.: Normal forms for certain singularities of vector fields. Ann. Inst. Fourier 23 (1973), 2.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Sotomayor, J., Garcia, R.A. Lines of curvature near principal cycles. Ann Glob Anal Geom 10, 275–289 (1992). https://doi.org/10.1007/BF00136870
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00136870