Planar vector field versions of Carathéodory's and Loewner's conjectures.

Título inglés	Planar vector field versions of Carathéodory's and Loewner's conjectures.
Título español	Versiones de campo vectorial plano de las conjeturas de Carathéodory y Loewner.
Autor/es	Gutiérrez, Carlos ; Sánchez Bringas, Federico
Organización	IMPA, Rio de Janeiro, Brasil;Dep. Mat. Fac. Cienc. Univ. Nac. Autón. México (UNAM), México D.F., Méjico
Revista	0214-1493
Publicación	1997, 41 (1): 169-179, 22 Ref.
Tipo de documento	articulo
Idioma	Inglés
Resumen inglés	Let r = 3, 4, ... , ∞, ω. The Cr-Carathéodory's Conjecture states that every Cr convex embedding of a 2-sphere into R3 must have at least two umbilics. The Cr-Loewner's conjecture (stronger than the one of Carathéodory) states that there are no umbilics of index bigger than one. We show that these two conjectures are equivalent to others about planar vector fields. For instance, if r ≠ ω, Cr-Carathéodory's Conjecture is equivalent to the following one: Let ρ > 0 and β: U Ì R2 → R, be of class Cr, where U is a neighborhood of the compact disc D(0, ρ) Ì R2 of radius ρ centered at 0. If β restricted to a neighborhood of the circle ∂D(0, ρ) has the form β(x, y) = (ax2 + by2)/(x2 + y2), where a < b < 0, then the vector field (defined in U) that takes (x, y) to (βxx(x, y) - βyy(x, y), 2βxy(x, y)) has at least two singularities in D(0, ρ).
Clasificación UNESCO	120219
Palabras clave español	Formas cuadráticas ; Campos vectoriales ; Sistemas diferenciales ; Puntos singulares ; Umbilicus
Código MathReviews	MR1461649
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