DOCSLIB.ORG

Deriving the Shape of Surfaces from Its Gaussian Curvature

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Deriving the Shape of Surfaces from Its Gaussian Curvature EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2019 Deriving the shape of surfaces from its Gaussian curvature FELIX ERKSELL SIMON LENTZ KTH SKOLAN FÖR TEKNIKVETENSKAP DEGREE PROJECT IN TECHNOLOGY, FIRST CYCLE, 15 CREDITS STOCKHOLM, SWEDEN 2019 Deriving the shape of surfaces from its Gaussian curvature FELIX ERKSELL SIMON LENTZ KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES Contents 1 Introduction 1 2 Surfaces 2 2.1 Notation . 2 2.2 Regularsurfaces ...................................... 3 2.3 Reparameterization . 3 2.4 Curves............................................ 4 2.5 Tangentplanes ....................................... 6 2.6 Derivatives ......................................... 7 3 Curvature 10 3.1 Intrinsicandextrinsicgeometry. 10 3.2 Thefirstfundamentalform ................................ 10 3.3 The second fundamental form . 12 3.4 TheGaussmap....................................... 12 3.5 TheWeingartenmap.................................... 14 3.6 TheGaussiancurvature .................................. 16 3.6.1 Normal curvature . 16 3.7 Umbilics........................................... 18 3.8 Weingarten equations . 18 3.9 Theprincipalcurvatures.................................. 20 4 Gauss Theorema Egregium 23 4.1 The Christo↵elsymbols .................................. 23 4.2 A note on notation . 24 4.3 Proof of the Theorema Egregium . 25 4.4 The Codazzi-Mainardi equations . 26 4.5 Principalpatches...................................... 26 4.5.1 The Christo↵elsymbols .............................. 27 4.5.2 The Gaussian curvature . 27 4.5.3 The Codazzi-Mainardi equations and the principal curvatures . 28 5 The Final Result 29 5.1 Statementoffinalresult .................................. 29 5.2 Proofofthetheorem.................................... 31 6 End note 33 1 Abstract A global statement about a compact surface with constant Gaussian curvature is derived by elemen- tary di↵erential geometry methods. Surfaces and curves embedded in three-dimensional Euclidian space are introduced, as well as several key properties such as the tangent plane, the first and sec- ond fundamental form, and the Weingarten map. Furthermore, intrinsic and extrinsic properties of surfaces are analyzed, and the Gaussian curvature, originally derived as an extrinsic property, is proven to be an intrinsic property in Gauss Theorema Egregium. Lastly, through the aid of umbil- ical points on a surface, the statement that a compact, connected surface with constant Gaussian curvature is a sphere is proven. Sammanfattning Ett globalt resultat f¨oren kompakt yta med konstant Gausskr¨okning h¨arleds med hj¨alp av grund- l¨aggande di↵erentialgeometri f¨orytor. Ytor och kurvor inb¨addade i det tredimensionella euklidiska rummet, tillsammans med centrala koncept som tangentplan, den f¨orsta- och andra fundamentala formen, och Weingartenavbildningen, introduceras. Vidare analyseras intrinsiska och extrinsiska egenskaper hos ytor, och Gausskr¨okningen, som h¨arleds genom extrinsiska metoder, visas vara en intrinsisk egenskap genom Gauss Theorema Egregium. Avslutningsvis visas det centrala resultatet att en kompakt sammanh¨angande yta med konstant Gausskr¨okning ¨aren sf¨ar. Chapter 1 Introduction Simply put, the goal of this project is to prove the following theorem, which at first sight may not seem very interesting: If a surface is connected, compact and has constant Gaussian curvature, then the surface is a sphere. What makes this result, and this project, interesting is the concept of Gaussian curvature and the concepts involved in understanding it. Firstly, the theorem requires one to gain an understanding of the mathematics of surfaces, namely di↵erential geometry. Secondly, one has to understand what it means for a surface to be curved, and especially the concept of Gaussian curvature. Finally, to really appreciate the theorem, one has to gain an understanding of the concepts of extrinsic and intrinsic geometry. This project starts with a brief introduction to the subject of di↵erential geometry for surfaces. There, the tools necessary to understand what a surface is and how one describes them are devel- oped. These are then built upon in the second part, where the central concept is that of curvature. The second part begins with a discussion about intrinsic and extrinsic geometry and then defines the curvature of a surface by extrinsic methods. This means relating the surface and its properties to the exterior space in which the surface lies. Intrinsic properties, on the other hand, can briefly be described as properties that are independent of the fact that the surface is embedded in a higher dimensional space, and which can be determined entirely by measurements made on the surface. It is also in this section that the Gaussian curvature is defined. This is done entirely by extrinsic considerations, and the remarkable result that the Gaussian curvature in fact is an intrinsic property is then the subject of the next part, Chapter 4, where Gauss’ famous Theorema Egregium is stated and proved. The project then concludes with the result relating the Gaussian curvature of a surface to its shape, as stated above. 1 Chapter 2 Surfaces This chapter is dedicated to introducing surfaces formally. Following the definition, several prop- erties and results concerning surfaces will be stated and discussed. Before that however, a small section that deals with the notation that will be used in this project is needed. 2.1 Notation The concepts of parameterizations and coordinates in R2 and R3 are useful and heavily relied on throughout the project, thus it is convenient to define the key quantities in the beginning. Consistent use of this notation follows throughout the report. Coordinates in R2 will be denoted by u1 and u2, and coordinates in R3 will be denoted by x1, x2, x3. However, it is quite cumbersome to always denote either two or three di↵erent components when all directions are being considered. To avoid this, coordinates will be denoted not by separate components, but as a component with an index, e.g. ui denotes both u1 and u2 when i = 1 and when i = 2, respectively. For coordinates in R3 the same index convention will be used, but Greek letters replace the Latin letters, mainly to be able to separate the coordinates completely, thus x↵ denotes x1 when ↵ = 1, x2 when ↵ = 2, x3 when ↵ = 3. It is to be understood that Latin indices run over 1, 2whileGreekindicesrunover1, 2, 3. Simplifying even further, the Einstein summation convention will also be adopted. This states that when two indices are present in one term, the indices are summed over the corresponding values. An example that shows the efficiency of this notation follows; if x : R2 R3, such that ! ↵ i x = x (u )e↵, where e 3 is the standard Cartesian basis, then the derivative with respect to ui becomes { ↵}↵=1 @x @x↵(uj) = e := x . @ui @ui ↵ i In the right hand side of the above equation, the index ↵ appears twice in the same term, thus it is assumed that the ↵-index will be summed over from one to three. The same follows for higher order derivatives @x @x↵ x = = e . ij @uiuj @uiuj ↵ 2 Quantities with two indices can be represented as the components of a matrix, which will be denoted by using parentheses. For example, if Lij are some quantities, then the corresponding matrix will be denoted as (Lij). The last piece of notation that will be introduced here is the ”dot”-derivative notation. For a function that is parameterized by t such that t f(t), then the derivative of this function will be denoted by 7! df = f.˙ dt 2.2 Regular surfaces This section is central to the whole chapter, as a formal definition of surfaces is given. The idea behind surfaces is that a subset of R3 should be defined by two coordinates, taken from R2.There are, however, restrictions that follow when defining a surface. One such restriction is that when standing on the surface, a ”geometrical linearization” must be allowed, namely around a point on the surface it should be well approximated as a two-dimensional plane1. Definition 2.1. If U R2 and V R3, then U and V are said to homeomorphic if there exists a continuous bijection⇢ with a continuous⇢ inverse function between them. The bijection is called a homeomorphism2. This definition is useful for surfaces because it is desirable to be able to describe areas of the surface with two coordinates, which can be done with homeomorphisms. Definition 2.2. If U R2, where U is open, then x : U R3 is said to be smooth if x↵ has continuous partial derivatives⇢ of all orders3. ! 3 Definition 2.3. Asmoothmapx : U R is said to be regular if x1 and x2 are linearly indepen- ! dent. Furthermore, a map x : U R3 is called a surface patch. ! Finally, the definition of a surface is given. Definition 2.4. A subset R3 is a regular surface if for each point p there exists an open S⇢ 2S subset U R2, a neighbourhood V of p in R3, and a regular surface patch x : U V which is a homeomorphism.⇢ The subset V is called an open subset of 4. ! \S \S S The key idea behind this definition is that two parameters (ui) from R2 are mapped to a point ↵ 3 (x )inR , and the regularity of the surface ensures that the derivatives xi are defined everywhere for the surface. Intuitively, this definition can be understood as for every point on a surface , S there should exist a bijection between an open subset U R2 and an open subset V R3. ⇢ \S⇢ 2.3 Reparameterization Generally, there are several di↵erent ways to parameterize a surface. A point p on a surface can be in the image of an infinite amount of di↵erent surface patches. Imagine having two di↵erentS surface 1K¨uhnel, p. 55 2Pressley, p. 68 3Pressley, p. 76 4Pressley, p. 68 3 Figure 2.1: A surface patch x taking coordinates from U to in R3. S patches, x : U W and x˜ : U˜ W˜ , with the point p belonging to both the open subsets W and !W˜S,\ namely, p !WS\W˜ .
Load more

Recommended publications
  • An Introduction to Topology the Classification Theorem for Surfaces by E
    An Introduction to Topology An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman Introduction. The classification theorem is a beautiful example of geometric topology. Although it was discovered in the last century*, yet it manages to convey the spirit of present day research. The proof that we give here is elementary, and its is hoped more intuitive than that found in most textbooks, but in none the less rigorous. It is designed for readers who have never done any topology before. It is the sort of mathematics that could be taught in schools both to foster geometric intuition, and to counteract the present day alarming tendency to drop geometry. It is profound, and yet preserves a sense of fun. In Appendix 1 we explain how a deeper result can be proved if one has available the more sophisticated tools of analytic topology and algebraic topology. Examples. Before starting the theorem let us look at a few examples of surfaces. In any branch of mathematics it is always a good thing to start with examples, because they are the source of our intuition. All the following pictures are of surfaces in 3-dimensions. In example 1 by the word “sphere” we mean just the surface of the sphere, and not the inside. In fact in all the examples we mean just the surface and not the solid inside. 1. Sphere. 2. Torus (or inner tube). 3. Knotted torus. 4. Sphere with knotted torus bored through it. * Zeeman wrote this article in the mid-twentieth century. 1 An Introduction to Topology 5.
    [Show full text]
  • Surface Physics I and II
    Surface physics I and II Lectures: Mon 12-14 ,Wed 12-14 D116 Excercises: TBA Lecturer: Antti Kuronen, [email protected] Exercise assistant: Ane Lasa, [email protected] Course homepage: http://www.physics.helsinki.fi/courses/s/pintafysiikka/ Objectives ● To study properties of surfaces of solid materials. ● The relationship between the composition and morphology of the surface and its mechanical, chemical and electronic properties will be dealt with. ● Technologically important field of surface and thin film growth will also be covered. Surface physics I 2012: 1. Introduction 1 Surface physics I and II ● Course in two parts ● Surface physics I (SPI) (530202) ● Period III, 5 ECTS points ● Basics of surface physics ● Surface physics II (SPII) (530169) ● Period IV, 5 ECTS points ● 'Special' topics in surface science ● Surface and thin film growth ● Nanosystems ● Computational methods in surface science ● You can take only SPI or both SPI and SPII Surface physics I 2012: 1. Introduction 2 How to pass ● Both courses: ● Final exam 50% ● Exercises 50% ● Exercises ● Return by ● email to [email protected] or ● on paper to course box on the 2nd floor of Physicum ● Return by (TBA) Surface physics I 2012: 1. Introduction 3 Table of contents ● Surface physics I ● Introduction: What is a surface? Why is it important? Basic concepts. ● Surface structure: Thermodynamics of surfaces. Atomic and electronic structure. ● Experimental methods for surface characterization: Composition, morphology, electronic properties. ● Surface physics II ● Theoretical and computational methods in surface science: Analytical models, Monte Carlo and molecular dynamics sumilations. ● Surface growth: Adsorption, desorption, surface diffusion. ● Thin film growth: Homoepitaxy, heteroepitaxy, nanostructures.
    [Show full text]
  • DISCRETE DIFFERENTIAL GEOMETRY: an APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 15: CURVATURE
    DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 15: CURVATURE DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 Curvature—Overview • Intuitively, describes “how much a shape bends” – Extrinsic: how quickly does the tangent plane/normal change? – Intrinsic: how much do quantities differ from flat case? N T B Curvature—Overview • Driving force behind wide variety of physical phenomena – Objects want to reduce—or restore—their curvature – Even space and time are driven by curvature… Curvature—Overview • Gives a coordinate-invariant description of shape – fundamental theorems of plane curves, space curves, surfaces, … • Amazing fact: curvature gives you information about global topology! – “local-global theorems”: turning number, Gauss-Bonnet, … Curvature—Overview • Geometric algorithms: shape analysis, local descriptors, smoothing, … • Numerical simulation: elastic rods/shells, surface tension, … • Image processing algorithms: denoising, feature/contour detection, … Thürey et al 2010 Gaser et al Kass et al 1987 Grinspun et al 2003 Curvature of Curves Review: Curvature of a Plane Curve • Informally, curvature describes “how much a curve bends” • More formally, the curvature of an arc-length parameterized plane curve can be expressed as the rate of change in the tangent Equivalently: Here the angle brackets denote the usual dot product, i.e., . Review: Curvature and Torsion of a Space Curve •For a plane curve, curvature captured the notion of “bending” •For a space curve we also have torsion, which captures “twisting” Intuition: torsion is “out of plane bending” increasing torsion Review: Fundamental Theorem of Space Curves •The fundamental theorem of space curves tells that given the curvature κ and torsion τ of an arc-length parameterized space curve, we can recover the curve (up to rigid motion) •Formally: integrate the Frenet-Serret equations; intuitively: start drawing a curve, bend & twist at prescribed rate.
    [Show full text]
  • Surface Water
    Chapter 5 SURFACE WATER Surface water originates mostly from rainfall and is a mixture of surface run-off and ground water. It includes larges rivers, ponds and lakes, and the small upland streams which may originate from springs and collect the run-off from the watersheds. The quantity of run-off depends upon a large number of factors, the most important of which are the amount and intensity of rainfall, the climate and vegetation and, also, the geological, geographi- cal, and topographical features of the area under consideration. It varies widely, from about 20 % in arid and sandy areas where the rainfall is scarce to more than 50% in rocky regions in which the annual rainfall is heavy. Of the remaining portion of the rainfall. some of the water percolates into the ground (see "Ground water", page 57), and the rest is lost by evaporation, transpiration and absorption. The quality of surface water is governed by its content of living organisms and by the amounts of mineral and organic matter which it may have picked up in the course of its formation. As rain falls through the atmo- sphere, it collects dust and absorbs oxygen and carbon dioxide from the air. While flowing over the ground, surface water collects silt and particles of organic matter, some of which will ultimately go into solution. It also picks up more carbon dioxide from the vegetation and micro-organisms and bacteria from the topsoil and from decaying matter. On inhabited watersheds, pollution may include faecal material and pathogenic organisms, as well as other human and industrial wastes which have not been properly disposed of.
    [Show full text]
  • The Gaussian Curvature 13/09/2007 Renzo Mattioli
    The Gaussian Curvature 13/09/2007 Renzo Mattioli The Gaussian Curvature Map Renzo Mattioli (Optikon 2000, Roma*) *Author is a full time employee of Optikon 2000 Spa (Int.: C3) Historical background: Theorema Egregium by Carl Friderich Gauss Refractive On Line 2007 1 The Gaussian Curvature 13/09/2007 Renzo Mattioli Historical background: C.F. Gauss (1777-1855) was professor of mathematics in Göttingen (D). Has developed theories adopted in – Analysis – Number theory – Differential Geometry – Statistic (Guaussian distribution) – Physics (magnetism, electrostatics, optics, Astronomy) "Almost everything, which the mathematics of our century has brought forth in the way of original scientific ideas, attaches to the name of Gauss.“ Kronecker, L. Gaussian Curvature of a surface (Definition) Gaussian curvature is the product of steepest and flattest local curvatures, at each point. Negative GC Zero GC Positive GC Refractive On Line 2007 2 The Gaussian Curvature 13/09/2007 Renzo Mattioli Gauss’ Theorema Egregium (1828) Gaussian curvature (C1 x C2) of a flexible surface is invariant. (non-elastic isometric distortion) C1=0 C1=0 C2>0 C2=0 C1 x C2 = 0 C1 x C2 = 0 Gauss’ Theorema Egregium Gaussian curvature is an intrinsic property of surfaces C1 C2 C1 C2 Curvature Gaussian Curvature Gaussian C2 C2 C1 x C2 C1 x C2 C1 C1 Refractive On Line 2007 3 The Gaussian Curvature 13/09/2007 Renzo Mattioli Gauss’ Theorema Egregium • “A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion. The Mercator projection… preserves angles but fails to preserve area.” (Wikipedia) The Gaussian Map in Corneal Topography Refractive On Line 2007 4 The Gaussian Curvature 13/09/2007 Renzo Mattioli Proposed by B.Barsky et al.
    [Show full text]
  • 3D Modeling: Surfaces
    CS 430/536 Computer Graphics I Overview • 3D model representations 3D Modeling: • Mesh formats Surfaces • Bicubic surfaces • Bezier surfaces Week 8, Lecture 16 • Normals to surfaces David Breen, William Regli and Maxim Peysakhov • Direct surface rendering Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University 1 2 http://gicl.cs.drexel.edu 1994 Foley/VanDam/Finer/Huges/Phillips ICG 3D Modeling Representing 3D Objects • 3D Representations • Exact • Approximate – Wireframe models – Surface Models – Wireframe – Facet / Mesh – Solid Models – Parametric • Just surfaces – Meshes and Polygon soups – Voxel/Volume models Surface – Voxel – Decomposition-based – Solid Model • Volume info • Octrees, voxels • CSG • Modeling in 3D – Constructive Solid Geometry (CSG), • BRep Breps and feature-based • Implicit Solid Modeling 3 4 Negatives when Representing 3D Objects Representing 3D Objects • Exact • Approximate • Exact • Approximate – Complex data structures – Lossy – Precise model of – A discretization of – Expensive algorithms – Data structure sizes can object topology the 3D object – Wide variety of formats, get HUGE, if you want each with subtle nuances good fidelity – Mathematically – Use simple – Hard to acquire data – Easy to break (i.e. cracks represent all primitives to – Translation required for can appear) rendering – Not good for certain geometry model topology applications • Lots of interpolation and and geometry guess work 5 6 1 Positives when Exact Representations Representing 3D Objects • Exact
    [Show full text]
  • Differential Geometry: Curvature and Holonomy Austin Christian
    University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-5-2015 Differential Geometry: Curvature and Holonomy Austin Christian Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Mathematics Commons Recommended Citation Christian, Austin, "Differential Geometry: Curvature and Holonomy" (2015). Math Theses. Paper 5. http://hdl.handle.net/10950/266 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. DIFFERENTIAL GEOMETRY: CURVATURE AND HOLONOMY by AUSTIN CHRISTIAN A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics David Milan, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler May 2015 c Copyright by Austin Christian 2015 All rights reserved Acknowledgments There are a number of people that have contributed to this project, whether or not they were aware of their contribution. For taking me on as a student and learning differential geometry with me, I am deeply indebted to my advisor, David Milan. Without himself being a geometer, he has helped me to develop an invaluable intuition for the field, and the freedom he has afforded me to study things that I find interesting has given me ample room to grow. For introducing me to differential geometry in the first place, I owe a great deal of thanks to my undergraduate advisor, Robert Huff; our many fruitful conversations, mathematical and otherwise, con- tinue to affect my approach to mathematics.
    [Show full text]
  • Surface Topology
    2 Surface topology 2.1 Classification of surfaces In this second introductory chapter, we change direction completely. We dis- cuss the topological classification of surfaces, and outline one approach to a proof. Our treatment here is almost entirely informal; we do not even define precisely what we mean by a ‘surface’. (Definitions will be found in the following chapter.) However, with the aid of some more sophisticated technical language, it not too hard to turn our informal account into a precise proof. The reasons for including this material here are, first, that it gives a counterweight to the previous chapter: the two together illustrate two themes—complex analysis and topology—which run through the study of Riemann surfaces. And, second, that we are able to introduce some more advanced ideas that will be taken up later in the book. The statement of the classification of closed surfaces is probably well known to many readers. We write down two families of surfaces g, h for integers g ≥ 0, h ≥ 1. 2 2 The surface 0 is the 2-sphere S . The surface 1 is the 2-torus T .For g ≥ 2, we define the surface g by taking the ‘connected sum’ of g copies of the torus. In general, if X and Y are (connected) surfaces, the connected sum XY is a surface constructed as follows (Figure 2.1). We choose small discs DX in X and DY in Y and cut them out to get a pair of ‘surfaces-with- boundaries’, coresponding to the circle boundaries of DX and DY.
    [Show full text]
  • Lines of Curvature on Surfaces, Historical Comments and Recent Developments
    S˜ao Paulo Journal of Mathematical Sciences 2, 1 (2008), 99–143 Lines of Curvature on Surfaces, Historical Comments and Recent Developments Jorge Sotomayor Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Rua do Mat˜ao 1010, Cidade Universit´aria, CEP 05508-090, S˜ao Paulo, S.P., Brazil Ronaldo Garcia Instituto de Matem´atica e Estat´ıstica, Universidade Federal de Goi´as, CEP 74001-970, Caixa Postal 131, Goiˆania, GO, Brazil Abstract. This survey starts with the historical landmarks leading to the study of principal configurations on surfaces, their structural sta- bility 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 Poincar´ein 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 Dynam- ical 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. 1. Introduction The book on differential geometry of D. Struik [79], remarkable for its historical notes, contains key references to the classical works on principal curvature lines and their umbilic singularities due to L. Euler [8], G. Monge [61], C. Dupin [7], G. Darboux [6] and A. Gullstrand [39], among others (see The authors are fellows of CNPq and done this work under the project CNPq 473747/2006-5. The authors are grateful to L.
    [Show full text]
  • On the Patterns of Principal Curvature Lines Around a Curve of Umbilic Points
    Anais da Academia Brasileira de Ciências (2005) 77(1): 13–24 (Annals of the Brazilian Academy of Sciences) ISSN 0001-3765 www.scielo.br/aabc On the Patterns of Principal Curvature Lines around a Curve of Umbilic Points RONALDO GARCIA1 and JORGE SOTOMAYOR2 1Instituto de Matemática e Estatística, Universidade Federal de Goiás Caixa Postal 131 – 74001-970 Goiânia, GO, Brasil 2Instituto de Matemática e Estatística, Universidade de São Paulo Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brasil Manuscript received on June 15, 2004; accepted for publication on October 10, 2004; contributed by Jorge Sotomayor* ABSTRACT In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface. Key words: Umbilic point, principal curvature lines, principal cycles. 1 INTRODUCTION The study of umbilic points on surfaces and the patterns of principal curvature lines around them has attracted the attention of generation of mathematicians among whom can be named Monge, Darboux and Carathéodory. One aspect – concerning isolated umbilics – of the contributions of these authors, departing from Darboux (Darboux 1896), has been elaborated and extended in several directions by Garcia, Sotomayor and Gutierrez, among others. See (Gutierrez and Sotomayor, 1982, 1991, 1998), (Garcia and Sotomayor, 1997, 2000) and (Garcia et al. 2000, 2004) where additional references can be found. In (Carathéodory 1935) Carathéodory mentioned the interest of non isolated umbilics in generic surfaces pertinent to Geometric Optics. In a remarkably concise study he established that any local analytic regular arc of curve in R3 is a curve of umbilic points of a piece of analytic surface.
    [Show full text]
  • Manifolds and Riemannian Geometry Steinmetz Symposium
    Manifolds and Riemannian Geometry Steinmetz Symposium Daniel Resnick, Christina Tonnesen-Friedman (Advisor) 14 May 2021 2 What is Differential Geometry? Figure: Surfaces of different curvature. Differential geometry studies surfaces and how to distinguish between them| and other such objects unique to each type of surface. Daniel Resnick (Union College) Manifolds and Riemannian Geometry 14 May 2021 2 / 16 3 Defining a Manifold We begin by defining a manifold. Definition A topological space M is locally Euclidean of dimension n if every point p 2 M has a neighborhood U such that there is a homeomorphism φ from U to an open subset of Rn. We call the pair (U; φ : U ! Rn) a chart. Definition A topological space M is called a topological manifold if the space is Hausdorff, second countable, and locally Euclidean. Daniel Resnick (Union College) Manifolds and Riemannian Geometry 14 May 2021 3 / 16 4 The Riemann Metric Geometry deals with lengths, angles, and areas, and \measurements" of some kind, and we will encapsulate all of these into one object. Definition The tangent space Tp(M) is the vector space of every tangent vector to a point p 2 M. Definition (Important!) A Riemann metric on a manifold to each point p in M of an inner product h ; ip on the tangent space TpM; moreover, the assignment p 7! h ; i is required to be C 1 in the following sense: if X and Y are C 1 vector fields on M, then 1 p 7! hXp; Ypip is a C function on M.A Riemann manifold is a pair (M; h ; i) consisting of the manifold M together with the Riemann metric h ; i on M.
    [Show full text]
  • Riemannian Submanifolds: a Survey
    RIEMANNIAN SUBMANIFOLDS: A SURVEY BANG-YEN CHEN Contents Chapter 1. Introduction .............................. ...................6 Chapter 2. Nash’s embedding theorem and some related results .........9 2.1. Cartan-Janet’s theorem .......................... ...............10 2.2. Nash’s embedding theorem ......................... .............11 2.3. Isometric immersions with the smallest possible codimension . 8 2.4. Isometric immersions with prescribed Gaussian or Gauss-Kronecker curvature .......................................... ..................12 2.5. Isometric immersions with prescribed mean curvature. ...........13 Chapter 3. Fundamental theorems, basic notions and results ...........14 3.1. Fundamental equations ........................... ..............14 3.2. Fundamental theorems ............................ ..............15 3.3. Basic notions ................................... ................16 3.4. A general inequality ............................. ...............17 3.5. Product immersions .............................. .............. 19 3.6. A relationship between k-Ricci tensor and shape operator . 20 3.7. Completeness of curvature surfaces . ..............22 Chapter 4. Rigidity and reduction theorems . ..............24 4.1. Rigidity ....................................... .................24 4.2. A reduction theorem .............................. ..............25 Chapter 5. Minimal submanifolds ....................... ...............26 arXiv:1307.1875v1 [math.DG] 7 Jul 2013 5.1. First and second variational formulas
    [Show full text]