Positive quadratic differential forms and foliations with singularities on surfaces

Guíñez, Víctor

doi:10.1090/s0002-9947-1988-0961601-4

Cited by 30 publications

(33 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A proof for the general case of Darbouxian singular points of quadratic forms can be found in [11]. Moreover, the characteristic polynomial is P ω1 = zz n + zz n = zz(z n−1 + z n−1 ), which has n − 1 real simple roots.…”

Section: Phase Portraits Near Darbouxian Singular Pointsmentioning

confidence: 99%

“…The quadratic forms with this property are called positive and have been studied by many authors [2], [6], [11], [13], [14]. A positive quadratic differential form defines a pair of transverse foliations in the region of regular points.…”

Section: Introductionmentioning

confidence: 99%

“…A linear differential form (n = 1) is always totally real. In the case n = 2, a quadratic differential form is totally real if it is positive in the sense of [11]. Take local coordinates x, y defined on some open subset U ⊂ M and assume that ω is given by…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Isolated singularities of binary differential equations of degree $n$

Fukui¹,

Nuño–Ballesteros²

2012

Publicacions Matemàtiques

View full text Add to dashboard Cite

We study isolated singularities of binary differential equations of degree n which are totally real. This means that at any regular point, the associated algebraic equation of degree n has exactly n different real roots (this generalizes the so called positive quadratic differential forms when n = 2). We introduce the concept of index for isolated singularities and generalize Poincaré-Hopf theorem and Bendixson formula. Moreover, we give a classification of phase portraits of the n-web around a generic singular point. We show that there are only three types, which generalize the Darbouxian umbilics D 1 , D 2 and D 3 .

show abstract

Section: Phase Portraits Near Darbouxian Singular Pointsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Isolated singularities of binary differential equations of degree $n$

Fukui¹,

Nuño–Ballesteros²

2012

Publicacions Matemàtiques

View full text Add to dashboard Cite

show abstract

“…An IDE of degree 1 is always totally real. In the case n = 2, an IDE is totally real if it is positive in the sense of [14]. In [12], Fukui and Nuño-Ballesteros introduce the concept of index for totally real IDE and produced a classification of generic singularities of this type of equations.…”

Section: Proposition 22 ([3]mentioning

confidence: 99%

Index of an implicit differential equation

Challapa¹,

Ruas²

2010

Publicacions Matemàtiques

View full text Add to dashboard Cite

show abstract

“…(See [23] Proof. Obviously we cannot take a special principal co-ordinate system at an umbilic; this is the key problem!…”

Section: Curvature-bde'smentioning

confidence: 99%

Dupin indicatrices and families of curve congruences

Bruce

Tari²

2004

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

Positive quadratic differential forms and foliations with singularities on surfaces

Cited by 30 publications

References 7 publications

Isolated singularities of binary differential equations of degree $n$

Isolated singularities of binary differential equations of degree $n$

Index of an implicit differential equation

Dupin indicatrices and families of curve congruences

Contact Info

Product

Resources

About