Lectures on the Qualitative Theory of Curves and Surfaces of R3 (original) (raw)

An Introduction to Differential Geometry: The Theory of Surfaces

From a mathematical perspective, a surface is a generalization of a plane which does not necessarily require being flat, that is, the curvature is not necessarily zero. Often, a surface is defined by equations that are satisfied by some coordinates of its points. A surface may also be defined as the image, in some space of dimensions at least three, of a continuous function of two variables (some further conditions are required to insure that the image is not a curve). In this case, one says that one has a parametric surface, which is parametrized by these two variables, called parameters. Parametric equations of surfaces are often irregular at some points. This is formalized by the concept of manifold: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane. A parametric surface is the image of an open subset of the Euclidean plane by a continuous function, in a topological space, generally a Euclidean space of dimension at least three. The paper aims at giving an introduction to the theory of surfaces from differential geometry perspective.

The Differential Geometry of Curves

Differential geometry of curves studies the properties of curves and higher-dimensional curved spaces using tools from calculus and linear algebra. This study has two aspects: the classical differential geometry which started with the beginnings of calculus and the global differential geometry which is the study of the influence of the local properties on the behavior of the entire curve. The local properties involves the properties which depend only on the behavior of the curve in the neighborhood of a point. The methods which have shown themselves to be adequate in the study of such properties are the methods of differential calculus. Due to this, the curves considered in differential geometry will be defined by functions which can be differentiated a certain number of times. The other aspect is the so-called global differential geometry which study the influence of the local properties on the behavior of the entire curve or surface. This paper aims to give an advanced introduction to the theory of curves, and those that are curved in general.

Principles of Differential Geometry

The present text is a collection of notes about differential geometry prepared to some extent as part of tutorials about topics and applications related to tensor calculus. They can be regarded as continuation to the previous notes on tensor calculus [9, 10] as they are based on the materials and conventions given in those documents. They can be used as a reference for a first course on the subject or as part of a course on tensor calculus.

Differential Geometry: An Introduction to the Theory of Curves

Differential geometry is a discipline of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry. The theory of plane, curves and surfaces in the Euclidean space formed the basis for development of differential geometry during the 18th and the 19th century. The core idea of both differential geometry and modern geometrical dynamics lies under the concept of manifold. A manifold is an abstract mathematical space, which locally resembles the spaces described by Euclidean geometry, but which globally may have a more complicated structure. The purpose of this paper is to give an elaborate introduction to the theory of curves, and those are, in general, curved. Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by applying the concept of differential and integral calculus. The curves are represented in parametrized form and then their geometric properties and various quantities associated with them, such as curvature and arc length expressed via derivatives and integrals using the idea of vector calculus.

A New Approach to the Fundamental Theorem of Surface Theory

Archive for Rational Mechanics and Analysis, 2007

Let ω be a simply-connected open subset of R 2. Given two smooth enough fields of positive definite symmetric, and symmetric, matrices defined over ω, the well-known fundamental theorem of surface theory asserts that, if these fields satisfy the Gauss and Codazzi-Mainardi relations in ω, then there exists an immersion from ω into R 3 such that these fields are the first and second fundamental forms of the surface (ω) We revisit here this classical result by establishing that a new compatibility relation, shown to be necessary by C. Vallée and D. Fortuné in 1996 through the introduction, following an idea of G. Darboux, of a rotation field on a surface, is also sufficient for the existence of such an immersion. This approach also constitutes a first step toward the analysis of models for nonlinear elastic shells where the rotation field along the middle surface is considered as one of the primary unknowns. Résumé Soit ω un ouvert simplement connexe de R 2. Etant donné deux champs suffisamment réguliers définis dans ω, l'un de matrices symétriques définies positives et l'autre de matrices symétriques, le théorème fondamental de la théorie des surfaces affirme que, si ces deux champs satisfont les relations de Gauss et Codazzi-Mainardi dans ω, alors il existe une immersion de ω dans R 3 telle que ces champs soient les première et deuxième forme fondamentales de la surface (ω). On donne ici une autre approche de ce résultat classique, en montrant qu'une nouvelle relation de compatibilité, dont C. Vallée et D. Fortuné ont montré en 1996 la nécessité en suivant une idée de G. Darboux, estégalement suffisante pour l'existence d'une telle immersion. Cette approche constitueégalement un premier pas vers l'analyse de modèles de coques non linéairement elastiques où le champ de rotations le long de la surface moyenne est pris comme l'une des inconnues principales.

Lines of Curvature on Surfaces, Historical Comments and Recent Developments

The São Paulo Journal of Mathematical Sciences, 2008

This survey starts with the historical landmarks leading to the study of principal configurations on surfaces, their structural stability and further generalizations. Here it is pointed out that in the work of Monge, 1796, are found elements of the qualitative theory of differential equations (QTDE), founded by Poincaré in 1881. Here are also outlined a number of recent results developed after the assimilation into the subject of concepts and problems from the QTDE and Dynamical Systems, such as Structural Stability, Bifurcations and Genericity, among others, as well as extensions to higher dimensions. References to original works are given and open problems are proposed at the end of some sections. Contents 1. Introduction 2. Historical Landmarks 2.1. The Landmarks before Poincaré: Euler, Monge and Dupin 2.2. Poincaré and Darboux 2.3. Principal Configurations on Smooth Surfaces in R 3 3. Curvature Lines near Umbilic Points 3.1. Preliminaries on Umbilic Points 3.2. Umbilic Points of Codimension One 3.3. Umbilic Points of Codimension Two 3.4. Umbilic Points of Immersions with Constant Mean Curvature 3.5. Curvature Lines around Umbilic Curves 4. Curvature Lines in the Neighborhood of Critical Points 4.1. Curvature Lines around Whitney Umbrella Critical Points 4.2. Curvature Lines near Conic Critical Points 4.3. Ends of Surfaces Immersed with Constant Mean Curvature 5. Curvature Lines near Principal Cycles 6. Curvature Lines on Canal Surfaces 7. Curvature Lines near Umbilic Connections and Loops 8. Principal Configurations on Algebraic Surfaces in R 3 9. Axial Configurations on Surfaces Immersed in R 4 9.1. Differential equation for lines of axial curvature 10. Principal Configurations on Immersed Hypersurfaces in R 4 10.1. Curvature lines near Darbouxian partially umbilic curves 10.2. Curvature lines near hyperbolic principal cycles 11. Concluding Comments References

A NOTE OF DIFFERENTIAL GEOMETRY

This note is about the application of the Method of the Repère Mobile to the Ellipsoid of Reference in Geodesy using the symplectic approach.