Navegando por Autor "Gomes, Renato Teixeira"

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Um breve estudo sobre a curvatura média e o teorema de Aleksandrov
(2022-06-08) Lira, Yasmin Alves Sobrinho; Gomes, Renato Teixeira; http://lattes.cnpq.br/0570606157057337; http://lattes.cnpq.br/4862014205090674
When looking for the eigenvalues of the differential of the normal Gauss map dN, naturally arise in its characteristic polynomial two functions that are invariant by change of base of this operator: the determinant of the matrix of the normal Gauss map, called Gaussian curvature and the trace of this application. As this linear map is self-adjoint,there is an orthonormal basis in which its matrix is written diagonally in terms of the principal curvatures, and its determinant and trace are given by det(dN) = (−k1)(−k2) and its dash by tr(dN) = −(k1+k2). The negative half of the H = k1 + k2/2 is the so-called mean curvature, which was introduced by French mathematician Sophie Germain when studying a problem related to membrane vibrations. At this time, a problem proposed by Lagrange, which later received the name of Plateau’s problem, a Belgian physicist who carried out several experiments and in-depth studies on soap films around 1850, was, roughly speaking, to determine a surface that has the smallest area among those which have the edge given by a prescribed Jordan curve. It can be shown that such a surface has zero mean curvature at its regular points. Such surfaces are called minimal and are named after Lagrange. In this work we will make a brief study on mean curvature and minimal surfaces,demonstrating some results and presenting some examples of such surfaces. Furthermore, we will demonstrate Aleksandrov’s theorem which under certain assumptions says that the only compact surface with constant mean curvature in R3 is the sphere. For this, we will demonstrate this result with a different “machinery” the one used by Aleksandrov. We will follow R. Reilly’s approach in his article “Mean Curvature, the Laplacian and Soap Bubbles” which makes use of more basic knowledge of differential and integral calculus and surface theory for its demonstration.
Um breve estudo sobre a geometria diferencial de superfícies em R3
(2021-07-23) Santos, Túlio José de Souza; Gomes, Renato Teixeira; http://lattes.cnpq.br/0570606157057337; http://lattes.cnpq.br/5181696493328012
The purpose of this work is to make a brief study on the differential geometry of surfaces in R3, with the objective of demonstrating the Gauss-Bonnet theorem in its local and global version. This relevant result relates the geometry and topology of surfaces in R3 and has very interesting consequences. Through it, it is possible to give an answer to an ancient problem of determining whether Euclid’s fifth postulate is an axiom or a theorem. In fact, what is obtained is that there is no harm in denying the fifth postulate, that is, to suppose that there may be more than one or no parallel line to a line r passing through a point p outside of r. What is found are "brave new worlds"that have different geometries from the Euclidean one.
Um breve estudo sobre o transporte paralelo, geodésicas e a aplicação exponencial
(2023-09-15) Costa, Matheus Rabelo Viana da; Gomes, Renato Teixeira; http://lattes.cnpq.br/0570606157057337; http://lattes.cnpq.br/3078665075835586
Geodesics are curves on a regular surface that have the property of locally minimizing length, that is, if two points are close together, the curve that has the shortest length connecting these two points is a geodesic. They are roughly the "straight lines"of the surface, as they have a constant velocity vector norm, and are zero acceleration curves. We can arrive at these curves through the solution of a variational problem, or following the "path of Geometry"in which we define geodesics as a curve whose field of tangent vectors is parallel. The study of these curves on a surface leads us to the knowledge of several important geometric properties, in addition to the development of new machinery, such as special coordinate systems, for example, which facilitate the study of surfaces and help in the calculation of their important geometric structures. In this work we will make a brief study about parallel transport, Geodesics and the exponential map and its properties. We will study the notion of a covariant derivative, and how we parallel transport vectors along curves. With this idea of parallelism, we will define geodesics as a curve that has a field of parallel tangent vectors, we will study some properties of these curves and the geodesic curvature of curves on surfaces. Finally, we will study the exponential map, the normal coordinate system and the geodesic polar coordinate system and we will use this one to, among other things, show that geodesics have the property of locally minimizing the length.
O Teorema Egregium
(2024-02-29) Gomes, Heloisa Cardoso Barbosa; Gomes, Renato Teixeira; http://lattes.cnpq.br/0570606157057337; http://lattes.cnpq.br/8017333927762482
During the development of differential geometry around the 17th century, an old problem occupied the minds of mathematicians at the time, which was determining whether the so-called 5th postulate of Euclid was in fact a postulate or a theorem. This postulate, which had an equivalent version published in 1795 by John Playfair (1748–1819), says that: through a point outside a given straight line it is possible to draw a single straight line parallel to the given straight line". There were many attempts to "prove"the fifth postulate, all of which failed. The answer to this question was given years later by Gauss, Lobachevski and Bolyai. In their work Disquisitiones generales circa superficies curves, Gauss shows that the curvature K(p) of a surface at the point p, initially calculated through the determinant of the differential of dNp which depends on the socalled first and second fundamental forms, actually depends only on the coefficients of the first fundamental form and their derivatives, and can be calculated using a formula that bears his name, the so-called Gauss formula. As a consequence of this formula we have the so-called Egregium Theorem which states that the Gaussian curvature of a surface is an invariant intrinsic, that is, it does not depend on the environment the surface is in and consequently, it is invariant due to local isometries. This discovery is closely related to non-Euclidean geometries, since the geometry of a surface with non-zero curvature is non-Euclidean. A consequence of this fact is that the 5th postulate is in fact a postulate and not a theorem. In this work, we will study the concepts necessary to understand Gauss’s Egregium theorem and its demonstration, as well as some applications of this important result.