# Ronaldo GarciaUniversidade Federal de Goiás | UFG · Instituto de Matemática e Estatística (IME)

Ronaldo Garcia

Doctor in Sciences (Math.)

Professor (see also the other profile in ResearchGate: Ronaldo Alves Garcia)

## About

29

Publications

2,779

Reads

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160

Citations

Citations since 2017

Introduction

Differential Equations and Dynamical Systems

## Publications

Publications (29)

We describe some three-dozen curious phenomena manifested by parabolas inscribed or circumscribed about certain Poncelet triangle families. Despite their pirouetting motion, parabolas' focus, vertex, directrix, etc., will often sweep or envelop rather elementary loci such as lines, circles, or points. Most phenomena are unproven though supported by...

In this paper are studied the simplest qualitative properties of asymptotic lines of a plane field in Euclidean space. These lines are the integral curves of the null directions of the normal curvature of the plane field, on the closure of the hyperbolic region, where the Gaussian curvature is negative. When the plane field is completely integrable...

Resumo. Neste trabalho, discutiremos o conceito matemático de dobra-duras generalizadas e de Beloch, que consistem na operação de realizar uma única dobradura de papel ao longo de uma reta (eixo), levando dois pontos dados sobre duas retas também dadas. Aplicamos esta conceitu-ação para resolver equações cúbicas pelo método de Lill e estabelecemos...

Resumo. Neste artigo são obtidos resultados de con-tagem em geometria plana e espacial relacionados com configurações de retas no plano, grandes círculos na esfera e planos no espaço. Várias noções de configurações são introduzidas e usamos argumen-tos heurísticos e o princípio da indução finita para obtenção dos números de regiões, arestas e vérti...

Matemático de renome internacional, o Professor Maurício Matos Peixoto faleceu em 28 de abril de 2019, aos 98 anos. Ele deixou um legado incomensurável para a Matemática e, em especial, para as áreas de Sistemas Dinâmicos e Teoria Qualitativa das Equações Diferenciais.

In 1872, G. Darboux defined a family of curves on surfaces of R-3 which are preserved by the action of the Mobius group and share many properties with geodesics. Here, we characterize these curves under the view point of Lorentz geometry and prove that they are geodesics in a 3-dimensional sub-variety of a quadric Lambda(4) contained in the 5-dimen...

In 1872, Gaston Darboux defined a family of curves on surfaces in the 3-dimensional Euclidean space E³ which are preserved by the action of the Möbius group and share many properties with geodesics. Here, we study the Darboux curves from a dynamical viewpoint on special canal surfaces, quadrics and some Darboux cyclides. We also describe the generi...

In 1872, Gaston Darboux defined a family of curves on surfaces in the 3-dimensional Euclidean space E3 which are preserved by the action of the Möbius group and share many properties with geodesics. Here, we study the Darboux curves from a dynamical viewpoint on special canal surfaces, quadrics and some Darboux cyclides. We also describe the generi...

We study the global behavior of foliations of ellipsoids by curves making a constant angle with the lines of curvature.

This paper establishes the geometric structure of the lines of principal
curvature of a hypersurface immersed in ${\mathbb R}^4$ in a neighborhood of
the set $\mathcal{S}$ of its principal curvature singularities, consisting of
the points at which atF least two principal curvatures are equal.
Under generic conditions defined by appropriate transver...

The topological structure of the lines of principal curvature, the umbilic
and partially umbilic singularities of all tridimensional ellipsoids of
${\mathbb R}^4$ is described.

In this paper is given an example of a discrete family of minimal tori T2T2 immersed in S3S3 such that all their principal lines are dense. A relationship between the dynamical behavior of principal foliations and transcendental number theory is established.RésuméDans cet article est presentée un exemple dʼune famille de surfaces minimales (tori) T...

In 1872 G. Darboux defined a family of curves on surfaces of R^3 which are preserved by the action of the Mobius group and share many properties with geodesics. Here we characterize these curves under the view point of Lorentz geometry and prove some general properties and make them explicit them on simple surfaces, retrieving results of Pell (1900...

We give a geometrical description of the principal curvature lines on an immersed canal hypersurface,and study the holonomy
of the one-dimensional principal foliation that they define.

In this paper it will be shown that a special trefoil knot has at least two vertices, i.e., the torsion vanishes at two distinct
points. Also a natural generalization to special knots will be established. Moreover it will be considered a class of special
trefoil knots that have four or more vertices.

In this paper an expository account on singularities of reversible vector fields on manifolds and boundary singularities is presented. Also we present the bifurcation diagram of a boundary cusp of codimension three, i.e, a Bogdanov-Takens singular point in the boundary of the semi plane {(x, y)∈R
2:x>-0} whose topological unfolding is given by the...

Consider oriented surfaces immersed in $\mathbb R^3.$ Associated to them, here are studied pairs of transversal foliations with singularities, defined on the Elliptic region, where the Gaussian curvature $\mathcal K$, given by the product of the principal curvatures $k_1, k_2$ is positive. The leaves of the foliations are the lines of harmonic mean...

In this paper, -principal cycles of surfaces immersed in 4 are studied. In terms of geometric invariants, an integral expression for the first derivative of the Poincar mapping associated with a -principal cycle is obtained.

Resumo. São apresentados alguns problemas e resultados sobre quadriláteros e polígonos. Será mostrado que um polígono con-vexo com lados fixados delimita uma região deárea máxima se, e somente se, está inscrito num círculo. Para os quadriláteros são obtidos, explicitamente em função dos comprimentos dos lados, á area e o raio do círculo. Introdução...

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 Principa...

The following extension of the Carathéodory conjecture on umbilics is proposed and solved—in very special cases: Let N: →2 be the Gauss map of a C3 ovaloid ⊂3. Let be the set of umbilics q∈ such that N(q)=q. If A is a connected component of , then, each connected component of \A contains at least an umbilic point of .

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 Poincaré-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and i...

## Projects

Projects (3)

The NEXUS Mathematicæ Journal is a publication of the Institute of Mathematics and Statistics at the Federal University of Goiás (IME-UFG) and aims at reuniting texts for scientific publication within the areas of Pure and Applied Mathematics, Mathematical Education, Statistics and their related areas. The journal welcome scientific articles, be they articles off disclosure, reports of experience, summary or survey of an area/sub-area as well as reviews. It may also be included, in each volume, an interview with significantly contributing professionals of the areas contemplated by the journal, always carried out as a formal invitation from the editorial board.
To submit your paper to Nexus Mathematicæ Journal, please visit:
https://www.revistas.ufg.br/nexus/about/submissions
To more information about our journal and past volumes, visit:
https://www.revistas.ufg.br/nexus/

The aim of this project is study special curves (curvature lines, asymptotic lines, axial lines, etc) of immersed manifolds using tools of the qualitative theory of Ordinary Differential Equations.

Extend in one dimension, in domain and target spaces ,
results obtained with Gutiérrez in Asterisque,89-99: 185-215, 1982.