Geometric mean curvature lines on surfaces immersed in R3

Here are studied pairs of transversal foliations with singularities, defined on the Elliptic region (where the Gaussian curvature 1’C is positive) of an oriented surface immersed in ]R3. The leaves of the foliations are the lines of geometric mean curvature, along which the normal curvature is given by B/~, which is the geometric mean curvature of the principal curvatures ki , k2 of the immersion. The singularities of the foliations are the umbilic points and parabolic curves, where 1~1 = k2 and K = 0, respectively. ~ * ~ Requ le 30 mai 2002, accepte le 21 octobre 2002 (1) Instituto de Matematica e Estatistica, Universidade Federal de Goias, CEP 74001970, Caixa Postal 131, Goiania, GO, Brazil. ~ Instituto de Matematica e Estatistica, Universidade de Sao Paulo, Rua do Matao 1010, Cidade Universitaria, CEP 05508-090, Sao Paulo, S.P., Brazil. The first author was partially supported by FUNAPE/UFG. Both authors are fellows of CNPq. This work was done under the project PRONEX/FINEP/MCT Conv. 76.97.1080.00 Teoria Qualitativa das Equações Diferenciais Ordinarias and CNPq Grant 476886/2001-5. Here are determined the structurally stable patterns of geometric mean curvature lines near the umbilic points, parabolic curves and geometric mean curvature cycles, the periodic leaves of the foliations. The genericity of these patterns is established. This provides the three essential local ingredients to establish sufficient conditions, likely to be also necessary, for Geometric Mean Curvature Structural Stability. This study, outlined at the end of the paper, is a natural analog and complement for the Arithmetic Mean Curvature and Asymptotic Structural Stability of immersed surfaces studied previously by the authors [6], [7], [9].