Ronaldo Garcia

Inflection points and topology of surfaces in 4-space

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 Poincare-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.

Structural Stability of Asymptotic Lines on Surfaces Immersed in R3

In this paper are studied immersions of surfaces into to R3 whose nets of asymptotic lines are topologically undisturbed under small perturbations of the immersion. These immersions are called structurally asymptotic stable. Sufficient conditions to belong to this class are established here. These conditions focus on the stable patterns around parabolic points, parabolic separatrix connections, periodic asymptotic lines (including those that intercept the parabolic lines) as well the exclusion of recurrent asymptotic lines. The class of immersions that are structurally stable in this sense is open in the C5-topology.

Lines of axial curvature on surfaces immersed in R4

Abstract The main notions concerning umbilics and lines of principal curvature, traditionally studied on surfaces in R 3 , are extended in this paper to surfaces immersed in R 4 . Departing from the classical second fundamental form and the ellipse of curvature of the immersion of a surface into R 4 here are studied in detail the concepts of a) axiumbilic points, analogous to classical umbilics since at them the ellipse of curvature has equal axes, and b) periodic lines of axial curvature, called here axial cycles, corresponding both to principal and mean curvature cycles in the classical R 3 case. To any immersed surface in R 4 its axial configurations: the principal configuration and the mean curvature configuration are associated. For surfaces in R 3 , the first one reduces to the configuration by umbilics and principal lines, while the second one gives the configuration by umbilics and integral foliations of the mean curvature line fields. Also the notion of principal structural stability of immersions of surfaces into R 3 is extended to that of axial structural stability, for the case of surfaces in R 4 . Sufficient conditions for the axial structural stability are provided in terms of axiumbilics, axial cycles and the asymptotic behavior of all the other lines of axial curvature

Structurally stable configurations of lines of mean curvature and umbilic points on surfaces immersed in R3

In this paper we study the pairs of orthogonal foliations on oriented surfaces immersed in R3 whose singularities and leaves are, respectively, the umbilic points and the lines of normal mean curvature of the immersion. Along these lines the immersions bend in R3 according to their normal mean curvature. By analogy with the closely related Principal Curvature Configurations studied in [S-G], [GS2], whose lines produce the extremal normal curvature for the immersion, the pair of foliations by lines of normal mean curvature and umbilics, assembled together, are called Mean Curvature Configurations. This paper studies the stable and generic cases of umbilic points and mean curvature cycles, with their Poincare map. This provides two of the essential local ingredients to establish sufficient conditions for mean curvature structural stability, the analog of principal curvature structural stability, [S-G], [GS2].

BIFURCATIONS OF UMBILIC POINTS AND RELATED PRINCIPAL CYCLES

The simplest patterns of qualitative changes on the configurations of lines of principal curvature around umbilic points on surfaces whose immersions into ℝ3 depend smoothly on a real parameter (codimension one umbilic bifurcations) are described in this paper.Global effects, due to umbilic bifurcations, on these configurations such as the appearance and annihilation of periodic principal lines, called also principal cycles, are also studied here.

Lines of mean curvature on surfaces immersed inR3

AbstractAssociated to oriented surfaces immersed inR3, here are studied pairs of transversal foliations with singularities, defined on theElliptic region, where the Gaussian curvatureK, given by the product of the principal curvaturesk1,k2 of the immersion, is positive. Theleaves of the foliations are thelines of M-mean curvature, along which the normal curvature of the immersion is given by a functionM=M(k1, k2) ∈ [k1, k2], called aM- mean curvature, whose properties extend and unify those of thearithmetic κ=(k1+k2)/2, thegeometric

Harmonic mean curvature lines on surfaces immersed in \( \mathbb{R}^{3} \)

\sqrt \mathcal{K}

Closed Principal Lines of Surfaces Immersed in the Euclidean 4-Space

andharmonic K/κ=((1/k1+1/k2)/2)−1classical mean curvatures.Thesingularities of the foliations are theumbilic points andparabolic curves, wherek1=k2 andK=0, respectively. Here are determined the patterns ofM- mean curvature lines near theumbilic points, parabolic curves andM-mean curvature cycles (the periodic leaves of the foliations), which are structurally stable under small perturbations of the immersion. The genericity of these patterns is also established.These patterns provide the three essential local ingredients to establish sufficient conditions, likely to be also necessary, forM-Mean Curvature Structural Stability of immersed surfaces. This constitutes a natural unification and complement for the results obtained previously by the authors for theArithmetic, [11],Asymptotic, [10, 14],Geometric, [12] andHarmonic, [13],classical cases ofMean Curvature, Structural Stability.