Lines Of Curvature On Surfaces Immersed In ℝ4

Abstract

The differential equation of the lines of curvature for immersions of surfaces into ℝ4 is established. It is shown that, for a class of generic immersions of a surface into ℝ4 in the Cr-topology, r ≥ 4, all of the umbilic points are locally topologically stable. This type of umbilic points is described. © 1997, Sociedade Brasileira de Matemática.282233251Burnside, W.S., Panton, A.W., (1912) The Theory of Equations, , [B-P] Dover Publications, Inc. New YorkGuadalupe, I., Gutiérrez, C., Sotomayor, J., Tribuzy, R., Principal Lines on Surfaces Minimally Immersed in Constantly Curved 4-spaces (1987) Dynamical Systems and Bifurcation Theory, Pitman Research Notes in Mathematics Series, 160, pp. 91-120. , [GGST]Gutierrez, C., Sotomayor, J., Principal Lines on Surfaces Immersed with Constant Mean Curvature (1986) Trans, of the Ame. Math. Soc., 293 (2), pp. 751-766. , [G-S]Jacobowitz, H., The Gauss-Codazzi Equations (1982) Tensor, N., S., 39, pp. 15-22. , [Jac]Little, J.A., On Singularities of Submanifolds of a Higher Dimensional Euclidean Space (1969) Ann. Mat. Pura App., 83, pp. 261-335. , [Lit]Palis, J., De Melo, W., (1982) Geometric Theory of Dynamical Systems, , [M-P] Springer-VerlagRamírez-Galarza, A., Sánchez-Bringas, F., Lines of Curvature near Umbilic Points on Surfaces Immersed in ℝ4 (1995) Annals of Global Analysis and Geometry, 13, pp. 129-140. , [R-S]Spivak, M., (1979) A Comprehensive Introduction to Differential Geometry, 5. , [Spi] Publish or Perish Inc., Berkele