Structural Stability of Asymptotic Lines on Surfaces Immersed in R3

García, Ronaldo; Gutiérrez, Carlos; Sotomayor, Jorge

doi:10.1016/s0007-4497(99)00116-5

Cited by 31 publications

(44 citation statements)

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“…This extends to the harmonic mean curvature setting the main theorems on structural stability for the arithmetic and geometric mean curvature configurations and for the asymptotic configurations, proved in [13,14,15,17].…”

Section: Introductionmentioning

confidence: 82%

“…The openness of A i (M 2 ) it follows from the local structure of the harmonic mean curvature lines near the umbilic points of types H i , i = 1, 2, 3, near the harmonic mean curvature cycles and by the absence of umbilic harmonic mean curvature separatrix connections and the absence of recurrences. The equivalence can be performed by the method of canonical regions and their continuation as was done in [19,21] for principal lines, and in [17], for asymptotic lines.…”

Section: On Harmonic Mean Curvature Structural Stabilitymentioning

confidence: 99%

See 1 more Smart Citation

Geometric mean curvature lines on surfaces immersed in ${\bf R}^3$

García¹,

Sotomayor²

2002

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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Here are studied pairs of transversal foliations with singularities, defined on the Elliptic region (where the Gaussian curvature K is positive) of an oriented surface immersed in R 3 . The leaves of the foliations are the lines of geometric mean curvature, along which the normal curvature is given by √ K, which is the geometric mean curvature of the principal curvatures k 1 , k 2 of the immersion.The singularities of the foliations are the umbilic points and parabolic curves, where k 1 = k 2 and K = 0, respectively.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].

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Section: Introductionmentioning

confidence: 82%

Section: On Harmonic Mean Curvature Structural Stabilitymentioning

confidence: 99%

Geometric mean curvature lines on surfaces immersed in ${\bf R}^3$

García¹,

Sotomayor²

2002

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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“…4 right and left, respectively. The construction of the topological equivalence can be performed using the method of canonical regions, see [14] and [19].…”

Section: Affine Curvature Lines Near Points With Double Eigenvalues Omentioning

confidence: 99%

“…The function δ has principal part (accordingly to the Newton's polygon) given by: It is worthwhile to mention that δ p does not depend on q 12 , q 03 , q 31 , q 22 , q 13 and q 04 . We can check using symbolic algebraic manipulators that the determinant of the Hessian of δ P (u, v) = δ p ( √ u, v) is det(Hess(δ P )(0, 0) = 2 14 When δ has a A ± 3 at the origin, by composition of diffeomorphism in the source, δ can be rewritten locally as v 2 ± u 4 , see [1,Chapter 11,page 188]. When δ has a A − 3 singularity at the origin, the double ξ-direction set is locally homeomorphic to v 2 − u 4 = 0, a pair of parabolas.…”

Section: 2mentioning

confidence: 99%

Lines of Affine Principal Curvatures of Surfaces in 3-Space

2020

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In this work we study the affine principal lines of surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic and hyperbolic). We also describe the generic behavior of affine curvature lines in the neighborhood of points with double eigenvalues (but not umbilics) of the affine shape operator and parabolic points.2010 Mathematics Subject Classification. Primary 53A15; secondary 37C75, 34A09, 53C12 .

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“…The local behaviour of these equations have been studied by several authors (see, for example, [1]- [14], [17]- [31]) and, in particular, all codimension ≤ 1 singularities and their bifurcations have been classified. Global aspects of certain types of BDE's have also been studied ( [18], [29]). …”

Section: Introductionmentioning

confidence: 99%

Dupin indicatrices and families of curve congruences

Bruce

Tari²

2004

Trans. Amer. Math. Soc.

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Abstract. We study a number of natural families of binary differential equations (BDE's) on a smooth surface M in R 3 . One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and principal BDE's, another between the characteristic and principal BDE's. The locus of singular points of the members of these families determine curves on the surface. In these two cases they are the tangency points of the discriminant sets (given by a fixed ratio of principle curvatures) with the characteristic (resp. asymptotic) BDE.More generally, we consider a natural class of BDE's on such a surface M , and show how the pencil of BDE's joining certain pairs are related to a third BDE of the given class, the so-called polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDE's are intimately related.

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Structural Stability of Asymptotic Lines on Surfaces Immersed in R3

Cited by 31 publications

References 0 publications

Geometric mean curvature lines on surfaces immersed in ${\bf R}^3$

Geometric mean curvature lines on surfaces immersed in ${\bf R}^3$

Lines of Affine Principal Curvatures of Surfaces in 3-Space

Dupin indicatrices and families of curve congruences

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