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˜ .