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 eciency 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˜ . Since the surface patches are homeomorphisms, it is S\ S\ 2S\ \ 1 1 meaningful to consider the open subsets of U and U˜; V = x ( W W˜ ) and V˜ = x˜ ( W W˜ ). Restricted to these subsets, x and x˜ cover the same part of theS\ surface.\ It is thereforeS possible\ \ to define a map which transitions between these two open subsets. This map is called a transition map and is formally defined as 1 =x x˜ : V˜ V. ! This gives the possibility of transitioning for all points in V˜ to the corresponding subset V by

x˜(˜ui)=x((˜ui)) and the opposite direction is also possibly via the inverse transition map. Two results about the transition map will be stated without proof. Theorem 2.1. The transition maps of a regular surface are smooth5.

Theorem 2.2. If U and U˜ are open subsets of R2 and x : U R3 is a regular surface patch, and if ˜ !1 ˜ ˜ 3 :U U is a bijective smooth map with smooth inverse : U U. Then, x˜ = x :U R is a regular! surface patch6. ! ! With the conditions from the above theorem, the surface patch x˜ is said to be a reparameteri- zation of x. The transition map brings with it a quantity called the Jacobian which is

@u1 @u2 @u˜1 @u˜1 J() = @u1 @u2 . @u˜2 @u˜2 !

2.4 Curves

This section gives a short introduction to curves. This is necessary because a lot of the properties of a surface can be derived by looking at the behavior of curves lying on the surface.

5Pressley, p. 78. 6Pressley, p. 78

4 Definition 2.5. A parameterized curve is a continuously di↵erentiable map : I R3, where ! I =(a, b) R. A regular parameterized curve is a parameterized curve with ˙ = 0 everywhere7. ⇢ 6

Figure 2.2: A parameterized curve : I R3. !

Definition 2.6. If : I R3 is a parameterized curve, then (t) has unit speed if ˙ (t) =1 everywhere8. ! k k Just as it was possible to reparameterize a surface, it is also possible to reparameterize a curve. An important property for regular curves is that they always can be reparametrized to have unit speed. This result will be taken for granted, and if nothing else is stated, all curves in this project will be assumed to have unit speed. The next result concerns unit speed curves and is very useful.

Theorem 2.3. If : I R3 is a unit speed curve, then ˙ ¨ =0for all t I. ! · 2 Proof. d d 0= ˙ = (˙ ˙ )=¨ ˙ + ˙ ¨ =2¨ ˙ . dtk k dt · · · ·

The curvature of curves will prove useful when discussing the curvature of surfaces. A few key concepts in the curvature of surfaces depend directly on the curvature of curves lying on the surface. Therefore, the curvature of curves is introduced. Intuitively, the curvature is a measure of how much the curve deviates from a straight line. The natural choice of straight lines on a curve are its tangent lines at various points. So, it is desirable to know how much the tangent vector at a point t is di↵erent from a tangent vector a small step away from this point. Luckily, there is a quantity that measures this, namely the second derivative of the curve. Since the magnitude of the tangent vector to the curve is unity, the magnitude of the second derivative of the curve will measure how much the tangent vector changes direction.

Definition 2.7. If : I R3 is a unit speed curve, then the curvature  of is ! (t)= ¨(t) k k for all t I9. 2 7K¨uhnel, p. 8 8Pressley, p. 11 9Vas - Curves,p.7

5 This definition merely states how much the curve changes direction as the parameter t changes value10. It should be noted that this definition does not hold for curves with non unit speed. But since all curves can be reparameterized to have unit speed, all curves can be assumed to have unit speed and this definition then holds. This section is concluded by defining the arc length of a curve. This will prove valuable when introducing the intrinsic geometry of surfaces.

Definition 2.8. The length of a curve between the points (t0) and (t) is the function s(t) given by11 t s(t)= ˙ (u) du. k k Zt0 2.5 Tangent planes

The next piece of information introduced concerns surfaces and is called the tangent plane to a surface at a point. This is the generalization of what a tangent vector is to curve, and these tangent vectors to curves will define the tangent plane. To see this, consider a curve lying on a surface. At a point t0 such that (t0)=p, the curve will have a tangent vector ˙ (t0) which is also tangent to the surface. Definition 2.9. A tangent vector to a surface at a given point is the tangent vector to a curve on the surface at that point12. By collecting all the tangent vectors evaluated at this point p it is then possible to define the tangent plane. Definition 2.10. The set of all tangent vectors to at point p make up the tangent plane to the surface at point p, denoted T 13. S 2S pS This definition states that two tangent planes are di↵erent at two di↵erent points, and that a tangent plane is only meaningful at the point of evaluation. Continuing the construction, there is a natural choice of basis for the tangent plane.

Theorem 2.4. Let x : U R3 be a surface patch to and let p be a point in the image of x(ui), i ! S 3 where u U. The tangent plane to at p is the vector subspace Tp R spanned by x1 and x2. 2 S S⇢ Proof. To prove this, a vector that is tangent to the surface at point p needs to be written as a linear combination of x1 and x2, this will prove that xi is a basis to the tangent plane, which are already linearly independent since x is regular. To do{ this,} consider a parametrized curve (t)=x(ui(t)) i lying on the surface such that the curve passes the point p at t⇤,i.e. (t⇤)=x(u (t⇤)) = p. S d i i Taking the derivative of the curve with respect to t yields ˙ = dt (x(u (t)) =u ˙ xi.Whichshows that the tangent vector to a curve lies in the space spanned by xi. i i Next, a vector spanned by xi is of the form c xi,wherc are coecients. Consider the i i i parametrized curve again, = x(u (t)) = x(u0 + c t), lying on the surface and at t = t⇤ i i S i let = x(u0 + c t⇤)=p. Then the derivative of the curve evaluated at t = t⇤ yields ˙ = c xi, 10Vas - Curves,p.7 11Pressley, p. 10 12Pressley, p. 85 13Pressley, p. 85

6 which shows that every vector spanned by xi is the tangent vector to a curve lying on the surface at a point p14.

Thus, for a regular surface patch x, a tangent plane spanned by the first order derivatives xi can always be constructed to the surface for any point in the image of x. A natural quantity arises from tangent planes, namely the ”direction” of the tangent plane. The unit normal vector to a plane describes which way the plane is facing, and two linearly independent vectors on the plane are already known. Therefore, with the vectors xi, it is possible to define the unit normal vector to a tangent plane.

Definition 2.11. Let x : U R3 be a surface patch to , let p be in the image of x.Theunit normal vector to the surface at! point p is S x x N(p)= 1 ⇥ 2 x x k 1 ⇥ 2k 15 where the vectors xi are evaluated at point p . However, there are two di↵erent unit vectors that are normal to the tangent plane which di↵er only in the sign. Definition 2.11 expresses what will be called the standard normal vector to a surface. It is fully possible, however, to choose a di↵erent surface patch, x˜ : U˜ R3 such that p also lies in the image of x˜. Then the unit normal vector associated to this surface! patch would be related to the standard unit normal vector by

N˜ (p) = sgn(det(J()))N(p) where J() is the Jacobian mentioned in section 2.3.16 Furthermore, it is possible to speak of an orientable surface. If one collects all the surface patches for a surface such that the transition maps between the surface patches have a positive Jacobi determinant, thenS the surface is orientable. This is captured in the following definition. Definition 2.12. An orientable surface is a surface with a collection of surface patches covering the whole surface such that det(J()) > 0 where J(S) is the Jacobian of the transition map (cf. Section 2.3) between two surface patches belonging to the surface patch collection17. For orientable surfaces, one is e↵ectively allowed to make the unit normal vector field into a smooth map, which makes the surface into an oriented surface. Throughout the rest of this project, all surfaces will be assumed to be oriented, and thus having a well defined smooth unit normal vector field.

2.6 Derivatives

The properties of surfaces stated above are now enough to introduce the derivative of a smooth map between surfaces. The main motivation for defining the derivative is that it will be essential

14Pressley, p. 85-86 15Pressley, p. 89 16Pressley, p. 89-90 17Pressley p. 90

7 Figure 2.3: A tangent plane, Tp , and unit normal vector N to the surface defined at point p . S S 2S when deriving the Gaussian curvature of a surface, which is defined through the derivative of a smooth map. To begin with, if the surfaces and ˜ have the surface patches x : U R3 and x˜ : U˜ R3, S S ! 1 ! respectively, then the map f : ˜ is a smooth map if the following composition x˜ f x : U U˜ is smooth18. Take f : S!˜Sto be such a smooth map between the surfaces and˜.Just as! in the single-variable calculusS! case,S the derivative should measure the change in theS functionS f when the point of evaluation is changed slightly. Intuitively, if one evaluates f at point p , and then make a small change to p + p , the derivative should measure the change in f during2S this change. The smaller one chooses the2 magnitudeS of p to be, the closer p + p will be to p. If one chooses it to be vanishingly small, then the line connecting p and p + p will be tangent to the surface . One is then led to believe that the derivative of the function f will be a map between S ˜ 19 the two tangent spaces of the respective surfaces, namely Dpf : Tp Tf(p) . More formally, if v belongs to the tangent plane of the surface S!at a pointS p,thenthisisthe tangent vector to a curve on ,withp = (t ) for some t . TheS smooth function f will then S 0 0 map this curve to another curve lying on the surface ˜ such that ˜ = f . Furthermore, at t = t0, the vector v˜ will be a tangent vector to the curve S˜ at point f(p), and thus lie in the tangent ˜ plane Tf(p) . With the current notation, the formal definition of the derivative of a smooth map is stated. S Definition 2.13. The derivative of the smooth map f at point p is the map D f : T 2S p pS! T ˜ such that if v T then D f(v)=v˜20. f(p)S 2 pS p Note that this definition of the derivative only depends on the point p, the function f : ˜, and the tangent vector v. In order for this definition to make sense, it must be shown thatS it! doesS not depend on the curve that has v as a tangent vector at point p (there are infinitely many curves that fulfill that condition). Begin by taking a surface patch x : U R3 to and a point p that lies in the image of the surface patch. The function f : ˜ will then! act asS S!S f(x(ui)) = x˜(↵i(uj))

18Pressley, p. 83 19Pressley, p. 88 20Pressley, p. 87

8 where ↵i : U R are smooth functions, and x˜ : U˜ R3 is a surface patch for ˜. Taking a curve lying on the surface! ; (t)=x(ui(t)), and a curve!˜ lying on the surface ˜; S˜(t)=x˜(˜ui(t)) such S i i j S that f maps to ˜, namely f((t)) = f(x(u (t))) = x˜(↵ (u (t))). Then at point t0, chosen so that i (t0) coincides with the point p,thecurve will have a tangent vector ˙ (t0)=u ˙ (t0)xi,whichlies in the tangent plane to at p. Let the derivative of f act on this tangent vector, which results in S i ˙ i i j Dpf(˙ (t0)) = Dpf(˙u (t0)xi)=u˜ (t0)x˜i = ↵ju˙ (t0)x˜i

i @↵i where ↵j := @uj . The equation above shows no dependence on the curve mentioned earlier. The equation takes only account to the point p, the function f, and the tangent vector at p; ˙ (t0), which as mentioned earlier is not dependent on any curve as several curves on can have the same tangent vector at the same point. Lastly, this shows also that the derivativeS is a linear map and i 21 can be expressed by the matrix (↵j). This concludes the necessary properties of curves and surfaces needed in order to discuss the intrinsic and extrinsic properties of surfaces.

21Pressley, p. 86-87

9 Chapter 3

Curvature

This chapter begins with a discussion of intrinsic and extrinsic geometry and curvature and then introduces the main tools necessary to describe these two: the first and the second fundamental forms. Then, the Gauss map and the Weingarten map are introduced, which in turns leads to a definition of the central concept in this project – the Gaussian curvature.

3.1 Intrinsic and extrinsic geometry

As briefly mentioned in the introduction, a central concept in this project is that of intrinsic geometry, which is most easily understood by first discussing extrinsic geometry. Extrinsic properties of a surface can intuitively be understood as properties which depend on the fact that the surface lies in a higher dimensional space. For example, when saying that a surface is curved, this can naively be said to describe how much a surface deviates from being a plane, a comparison which clearly is dependent of the exterior space. Another example is when saying that a surface is oriented in space. Clearly, this is something that only can be seen from an exterior viewpoint, and nothing that a two dimensional inhabitant on the surface would be able to deduce. Intuitively, properties that in theory can be described and measured by inhabitants living on a surface are called intrinsic properties. For example, measuring distances and areas on the surface is something that an inhabitant on it would be able to do and properties derived from these kind of measurements are then intrinsic. In this project, only two dimensional surfaces in three dimensional space are looked at. Since measurements of distances in no way is restricted to two dimensional spaces however, the notion of intrinsic geometry opens up for generalizations of certain concepts to higher dimensions. One non-trivial example is curvature, which in Chapter 4 will be proven to be an intrinsic property.

3.2 The first fundamental form

As mentioned in the discussion about intrinsic geometry, the main point of interest is measuring the lengths of curves that lie on the surface. In general, the length of a curve is given by the integral ˙ (t) dt (3.1) k k Z

10 taken over the appropriate values for t and where ˙ is the usual euclidean norm. With this defintion, the quantity k k ds = ˙ (t) dt (3.2) k k can be seen as the length of the infinitesimal curve segment traced out as t varies by an amount dt. If lies on a surface , its tangent ˙ will at all points lie in the tangent space of the surface at that point. Therefore, theS measuring of distances on a surface is connected to measuring the lengths of tangent vectors to the surface. However, seeing the surface as embedded in R3, this is not any di↵erent from measuring lengths of ordinary vectors and the following definition comes naturally. Definition 3.1. Let p be a point on a surface . The first fundamental form is the symmetric bilinear form defined by S v, w p, = v w, h i S · where v w is the usual dot product1. · The first fundamental form is therefore nothing else than the dot product but pointwise restricted to tangent vectors on the surface. The brackets and subscripts will often be left out and the first fundamental form will simply be denoted by the usual dot. The name ”the first fundamental form” comes from the fact that the dot product is an inner product (a positive bilinear form), a fact that here will be taken for granted. Suppose again that is a curve on .Ifx is a parameterization of containing p,then S S (t)=x(ui(t)) and its tangent vector is given by i ˙ = xiu˙ . By (3.2), the squared distance between two points seperated by an infinitesimal distance is then given by ds2 =(˙ ˙ ) dt2 = g duiduj, · ij where g = x x . The expression g duiduj is often referred to as the first fundamental form, or ij i · j ij the metric form to emphasize its role in measuring distances. The coecients gij will be referred to as the coecients of the first fundamental form and used extensively in the project2. A convenient notation when working with these coecients explicitly is

g11 = E, g12 = g21 = F, g22 = G.

This will mainly be used from the end of this chapter and then throughout the rest of the report. The main reason for introducing this is that complicated expressions can be treated very nicely with index notation and then the gij notation is very useful. However, some explicit results will also be needed, and then the symmetry of g12 = g21 becomes useful and more obvious when writing them as F . Also, the notation in this report uses indices to denote derivatives and it is easier to use E1 than g11,1, especially when the expressions are complicated. This way of measuring distances on a surface is based on the concept of locally at each point approximating the surface by its tangent plane and measuring the distances of tangent vectors in this plane and the distance between two points x(ui) and x(ui + dui) on the surface is given by

1Pressley, p.122 2[5] Vas, p. 5-6

11 i j3 ds = gijdu du . The most important property of the first fundamental form is that it allows lengths on the surface to be measured and that it is unchanged under reparametrizations, which p follows from the properties of inner products. The first fundamental form can be determined entirely from measurements made on the surface and without referring to the exterior space in which the surface is embedded. Therefore, all quantities expressed using the coecients of the first fundamental form will be intrinsic, i.e. possible to determine entirely by measurements made on the surface and not depending on if and how the surface is embedded in R3. This will later be used to prove Gauss Theorema Egregium4. The remainder of this chapter focuses on the concept of curvature for surfaces and is based entirely on extrinsic considerations. In the next chapter, however, the tools and results derived here will be used to prove that the Gaussian curvature actually is an intrinsic measure of curvature, despite the extrinsic definitions in the following sections.

3.3 The second fundamental form

An intuitive concept of the curvature of a surface is to look at its deviation from being a plane at each point. Let be a surface and x a parameterization of the surface containing the point x(ui) Then, the separationS vector between x(ui) and a nearby point x(ui + dui) can be orthogonally decomposed with one component in the tangent plane and one component along the unit normal vector N. The component along N describes how much the surface deviates from being a plane around x(ui). Taylor expanding to the second order gives 1 x(ui + dui)=x(ui)+x dui + x duiduj, i 2 ij where the derivatives are evaluated at the point x(ui). Since x N = 0, this gives i · 1 x(ui + dui) x(ui) N = L duiduj, · 2 ij where L = x N. In Section 3.5, it is shown that these coecients L can be obtained by a ij ij · ij symmetric bilinear form , : Tp Tp R, called the second fundamental form, as Lij = hh ii S⇥ S! xi, xj . This captures the extrinsic behavior of the surface around a given point and as for the firsthh fundamentalii form, the following notation will prove useful:

L11 = e, L12 = L21 = f, L22 = g.

A formal definition will be given in Section 3.5, where it also will be shown that the second fun- damental form is independent of the parameterization. The following two sections provide an alternative way to describe the curvature of a surface and will ultimately also lead to the second fundamental form.

3.4 The Gauss map

The previous section described the curvature of a surface by looking at how much the surface deviates from being a tangent plane around a point. A similar approach is to look at how the

3Alexandrov, p. 9 4[5] Vas, p.7

12 Figure 3.1: Tangent plane deviation normal unit vector varies. If the normal is constant, then the surface must be a plane and its curvature should be zero. The more the unit normal varies, the more curved the surface should be. It was previously shown that a regular surface has a unit normal vector defined at all points. If x : U R3 is a parameterization for a surface, then a unit normal vector field on the surface is defined by! x x N = 1 ⇥ 2 . x x k 1 ⇥ 2k This vector field is di↵erentiable since x is smooth and x1 x2 is non-zero because the surface is assumed to be regular. ⇥ By imagining an arbitrary surface, the normal vectors will point in arbitrary directions. However, if one were to put all these vectors in one common origin, all the vectors would lie on a sphere, where the di↵erent directions of the normal vectors indicate how the surface varies between di↵erent points. Thus, the unit normal vector field can be seen a map from the surface to the unit sphere in R3. This is all combined in the following definition.

Figure 3.2: The Gauss map

Definition 3.2. For an orientable surface with unit normal vector field N,themap S 2 2 3 N : S ,S R , S! ⇢

13 where S2 is the unit sphere in R3,iscalledtheGaussmapof 5. S However, as mentioned, what is interesting is how the normal vector changes around a given point. This is captured by the derivative of the Gauss map - the Weingarten map - which is the subject of the next section.

3.5 The Weingarten map

The curvature of a curve is given by a scalar which simply measures the amount by which a curve deviates from being a straight line as one travels along the curve. A similar approach for surfaces is to look at how the surface deviates from being a plane as one travels around a point. This can be done by using the Gauss map. Since the normal vector is closely related to the tangent space, by looking at how the normal varies, one can measure how the surface deviates from being a plane. For a surface however, the normal may vary di↵erently as one travels through a given point in di↵erent directions and it is therefore necessary to define the derivative as the derivative with respect to the parameter of the normal vector field when restricted to a curve on the surface. The derivative of a map between two surfaces is a map between the corresponding tangent spaces, see section 2.6. But the tangent space of a surface is determined by the normal vector of the surface, and since the Gauss map simply maps the unit normal to the unit sphere, then the normal vector will have the same direction in the sphere and on the surface, therefore the tangent plane of the sphere and the surface will be identical. Therefore, the derivative of the Gauss map can be seen as a linear operator on the tangent space of a surface. This derivative is called the Weingarten map and is defined below. 6 Definition 3.3. The Weingarten map of at p is the map p, : Tp Tp given by S 2S W S S! S p, = DpN. W S

Figure 3.3: The unit normal vector field restricted to a curve

One key property of the Weingarten map which later will prove central is the following.

5do Carmo, p. 136 6Pressley p. 163

14 Theorem 3.1. The Weingarten map : T T is a self-adjoint operator7. W pS! pS Proof. Let x be a parametrization of containing p such that x(ui )=p. It has to be proven that S 0 (v), w = v, (w) hW i h W i for two arbitrary tangent vectors v, w Tp . However, since x1, x2 is a basis for Tp and is linear, it suces to show that 2 S { } S W (x ), x = x , (x ) . (3.3) hW 1 2i h 1 W 2 i By definition, the derivative of the normal vector along the i:th coordinate line is @N (x )= D N(x )= = N W i p i @ui i and (3.3) can then be rewritten as N , x = x , N . (3.4) h 1 2i h 1 2i But this equality follows directly from di↵erentiating the identity N, x =0withrespecttouj: h ii @ 0= N, x = N , x + N, x @uj h ii h j ii h iji from which it follows that N , x = N, x . (3.5) h j ii h iji Since xij = xji and since the first fundamental form is symmetric, (3.4) follows. Since the Weingarten map is self-adjoint, it is possible to associate with it a symmetric bilinear form8. This is the formal definition of the second fundamental form. Definition 3.4. The second fundamental form of at p is a bilinear form that associates with two tangent vectors v, w T the scalar9 S 2S 2 pS v, w p, = p, (v), w p, . hh ii S hW S i S An immediate consequence of this definition and (3.5) is this special case

L = x , x = (x ), x = N , x = N, x , (3.6) ij hh i jii hW i ji h i ji h iji which shows that the definition of Lij in Section 3.3 is meaningful, as this shows that it is a bilinear symmetric form. The key property of self-adjoint operators that is needed in this project is contained in the following theorem, which is stated without proof. Theorem 3.2. Let A : V V , where V is a finite dimensional vector space, be a linear self-adjoint ! operator. Then there exists an orthonormal basis of vectors, ei , such that A(ei)=iei, where the scalars are the eigenvalues of A.Inthebasis e ,thematrixof{ } A is diagonal and the elements i { i} i, where 1 2, on the diagonal are the maximum and minimum, respectively, of the quadratic form defined by Q(v)= Av, v restricted to unit vectors in V 10. h i 7Gluck, p. 18 8Pressley, p. 380 9Pressley, p. 163 10do Carmo, p. 216

15 Since the Weingarten map is self-adjoint, this states that there exists an orthonormal basis for the tangent space and that in this basis, the matrix of the Weingarten map is diagonal. Further, it states that the eigenvalues corresponding to these eigenvectors are the maximum and minimum of the second fundamental form, restricted to unit vectors. The geometric meaning of this will be explained in the following section.

3.6 The Gaussian curvature

The Weingarten map was defined by looking at how the normal varies around a given point on a surface. Since the normal is related to the tangent space, this map described the curvature of the surface at that point. However, it is more convenient to have a scalar value than a map to measure the curvature and therefore the following definition is useful.

Definition 3.5. Let p and let : Tp Tp be the Weingarten map. The determinant of is the Gaussian curvature2S K of Wat p11S. ! S W S Since the Weingarten map is self-adjoint, there exists an orthonormal basis consisting of eigen- vectors to the Weingarten map for the tangent space. The eigenvectors are in this context often called the principal vectors and the eigenvalues are the corresponding principal curvatures, for rea- sons which will become clear later. The directions corresponding to the principal vectors are called the principal directions. In this basis, chosen in such a way that the principal curvatures fulfill 1 2, the matrix of takes the simple form W 1 0 0  ✓ 2◆ and the Gaussian curvature is given by K = 12. (3.7) This is important because it relates the Gaussian curvature to the two eigenvalues of two orthogonal directions on the surface at a given point. This has a very clear geometric interpretation which depends on the concept of normal curvature of curves.

3.6.1 Normal curvature If is a unit speed curve on a surface , its tangent vector at a given point will lie in the cor- responding tangent space. The second derivativeS ¨ will however in general point in an arbitrary direction in R3. The projection of ¨ on to the normal vector is called the normal curvature of at that point. It is clear that if is contained entirely in a plane, then the magnitude of the normal curvature is equal to the curvature of at that point. The central result about normal curvature is contained in the following theorem. Theorem 3.3. The normal curvatures of all curves on the surface at a point p with the same tangent vector are the same. If is a unit-speed curve, its normalS curvature is2 givenS by12,13

 = ˙ , ˙ . n hh ii 11do Carmo, p. 146 12do Carmo, p. 142 13Pressley, p. 167

16 Proof. Since N ˙ = 0, di↵erentiating gives N˙ ˙ + N ¨ = 0 and thus · · ·  = N ¨ = N˙ ˙ = (˙ ), ˙ = ˙ , ˙ . n · · hW i hh ii

This makes it possible to speak of the normal curvature along a given direction and also justifies the name principal curvatures for the eigenvalues of the Weingarten map. To see this, let ei be the i:th principal vector. Then

e , e = (e ), e =  e , e =  e , e =  , hh i iii hW i ii h i i ii ih i ii i where i is the corresponding principal curvature. Thus, the principal curvatures are the normal curvatures corresponding to the principal directions given by the Weingarten map. A nice visual representation of this is given by what is called normal sections.

Figure 3.4: A normal section

At each point on a surface, a plane is spanned by the normal N and any given tangent vector w. This plane intersects the surface along a curve called a normal section. By Theorem 3.3, the curvature of this curve is in magnitude given by w, w . The curvatures of the normal sections corresponding to the principal directions are thenhh givenii by the the principal curvatures at that point. Furthermore, these normal sections intersect at a right angle and Theorem 3.2 states that the curvature of these sections are the maximum and minimum of all possible curves passing that point. Thus, the Gaussian curvature at a given point can be seen as the product of the curvatures of the two curves with maximum and minimum curvature at that point, and these two curves intersect each other at a right angle. There is one case when this does not hold in general. This is when the principal curvatures are equal. Then all directions are principal directions (see the next section) and there is nothing special about curves that intersect orthogonally. However, such a pair of normal sections can always be chosen. The next section will discuss these points in more detail and also prove a result that later will be central in proving the final theorem in Chapter 5.

17 3.7 Umbilics

Points at which the principal curvatures are equal are called umbilic points, or umbilics. These can be understood as points where all normal sections have the same curvature. A simple example are the points on a plane. Through all such points, all normal sections will be straight lines with zero curvature. Another example is given by the following theorem, which is central to this project. Theorem 3.4. Let be a connected surface of which every point is an umbilic, and with non-zero principal curvature.S Then, is an open subset of a sphere. S Proof. Let x : U R3,whereU is a connected open subset of R2, be a surface patch of .If is ! S the principal curvature at some point, then (xi)=xi, which holds since the point is an umbilic. But (x )= N and therefore N = xW.Ifi = j,then W i i i i 6 (x ) = N = N =(x ) . i j ij ji j i

Since xij = xji, this gives @ @ x = x . @uj i @ui j i j But xi and xj are linearly independent, so @/@u = @/@u = 0, which shows that  is constant on . S Now, integrating Ni = xi gives N = x + a,wherea is some constant vector of integration. This finally yields 1 2 1 2 1 x a = N = ,   2 which shows that any surface patch x of is an open subset of a sphere centered at a/ with radius 1/. To show then that the entire surfaceS is an open subset of a sphere, it suces to note that if two patches overlap, since the  is constantS everywhere, they are both parts of the same sphere and that this must hold for all patches. Therefore, the entire surface is an open subset of asphere14. S The next section introduces some concepts that will be used in the next chapter to prove that the Gaussian curvature is in fact an intrinsic measure of curvature.

3.8 Weingarten equations

Let N be the unit normal vector of a surface at a point p and let x be a parameterization of containing p.If(t)=x(ui(t)) is a curve on S passing through p, then the derivative of N alongS is given by S

i i i DpN(˙ )=DpN(˙u xi)=N˙ (u )=u ˙ Ni. (3.8) Since the derivative of the Gauss map is a linear operator on the tangent space of the surface, N T . Therefore, there exists coecients aj such that i 2 pS i j Ni = ai xj. (3.9) 14Pressley p. 191-192

18 Equation (3.8) shows that the matrix representation of DpN in the xi basis is given by

a1 a1 1 2 . (3.10) a2 a2 ✓ 1 2◆

From (3.5), the coecients of the second fundamental form are given by Lij = Ni xj so equation (3.9) yields · L = al g (3.11) ij j li which is a multiplication of two matrices written in index notation. By introducing gij as the ir r r elements of the inverse matrix of (gij) and then using that glig = l ,wherel is the Kronecker delta, equation (3.11) gives L gir = al g gir = al r = ar, (3.12) ij j li j l j where the last equation follows from the definition of the Kronecker delta. The equations (3.9) with the coecients given by (3.12) are known as the Weingarten equations. In general, the inverse matrix of (gij) is given by

ij 1 1 (g )=(gij) = adj[(gij)], (3.13) det [(gij)] where adj[(g )] is the adjugate matrix of (g ). If (g ) is written as a 2 2 matrix ij ij ij ⇥ g g (g )= 11 12 , (3.14) ij g g ✓ 21 22◆ then its adjugate is g g adj[(g )] = 22 12 . (3.15) ij g g ✓ 21 11 ◆ j Using the above, the coecients ai can be calculated: fF eG a1 = , 1 EG F 2 eF fE a2 = , 1 EG F 2 (3.16) gF fG a1 = , 2 EG F 2 fF gE a2 = . 2 EG F 2 Also, since a1 a1 1 2 (3.17) a2 a2 ✓ 1 2◆ is the matrix for DpN in the xi basis, and since the determinant is the same as for the Weingarten map = D N, the gaussian curvature K can be expressed as W p eg f 2 K = a1a2 a1a2 = . (3.18) 1 2 2 1 EG F 2 19 To show that this is well defined, i.e. that EG F 2 = 0, consider 6 x x 2 =(x x ) (x x )=(x x )(x x ) (x x )(x x )= k 1 ⇥ 2k 1 ⇥ 2 · 1 ⇥ 2 1 · 1 2 · 2 1 · 2 2 · 1 = EG F 2, where the second equality uses the vector algebra identity

(a b) (c d)=(a c)(b d) (a d)(b c). ⇥ · ⇥ · · · · Since all surfaces are considered regular, x x is non-zero and thus EG F 2 is non-zero15,16. 1 ⇥ 2 3.9 The principal curvatures

The final section of this chapter states some results about the principal curvatures which will be needed in Chapter 5. The first theorem will be needed below to prove Theorem 3.6, an important theorem which states how the principal curvatures behave during reparametrizations.

3 Theorem 3.5. Let x : U R be a surface patch with gij = xi, xj and Lij = xi, xj . Then the principle curvatures are! given by the following equation:17 h i hh ii

det [(L g )] = 0. (3.19) ij ij

Proof. The Weingarten map is the negative of the derivative of DpN. Using (3.12) then shows that the elements r of the matrix of in the x basis are given by Wj W i r = L gir = L gri = griL . Wj ij ij ij By now denoting the matrices by (Lij)=L, (gij)=g and being careful not to identify g with anything else, the above can be written in matrix form as

r 1 = g L Wj and the principal curvatures are given as the roots to

1 det g L I =0, where I is the identity matrix. This can then be rewritten as

1 det(L g)) 0=det g (L g) = , det(g) which is well defined since det(g)=(EG F 2) is non-zero, by the previous discussion. This proves the theorem. 15Pressley, p. 140-141 16do Carmo, p. 154-155 17Pressley, p. 190

20 Theorem 3.6. Let x : U R3 be a surface patch. Then, the principal curvatures either stay the same or both change sign when! x is reparameterized18.

Proof. Let x : U R3 and ˜x : U˜ R3 be surface patches of the surface ,where˜x is a reparameterization! of the surface, as described! in section 2.3. Let ˜x haveg ˜ =S x˜ , x˜ , L˜ = ij h i ji ij x˜i, x˜j , and similarly, x has gij = xi, xj and Lij = xi, xj . From a basis change it is clear thathh theii following relation will hold h i hh ii @ui @uj g˜ = g . kl ij @u˜k @u˜l Identifying the partial derivatives in the expression above as the components of the Jacobian matrix from Section 2.5, then it is possible to write the above equation in matrix notation:

T (˜gkl)=J (gij)J.

The exact same calculations for the second fundamental form yield the similar expression:

(L˜ )= J T (L )J kl ± ij where the factor depends on the sign of the determinant of the Jacobian, namely a negative sign if the determinant± is negative, and a positive sign if the determinant is positive. Now, assume that the reparameterization ˜x gives the principal curvature ˜, which is obtained through Theorem 3.5

det (L˜ ˜g˜ ) =0. ij ij h i Consider the matrix of which the determinant is being taken of

(L˜ ˜g˜ )=(L˜ ) ˜(˜g )= J T (L )J ˜J T (g )J = J T [( L ˜g )]J. ij ij ij ij ± ij ij ± ij ij Consider first the situation when the determinant of the Jacobian is positive, namely if

(L˜ ˜g˜ )=J T [(L ˜g )]J ij ij ij ij and taking the determinant of this matrix gives

det (L˜ ˜g˜ ) =det(J)2 det[(L ˜g )] = 0 ij ij ij ij h i which gives the same equation that determines the principal curvatures for the surface patch x,thus the principal curvatures has the same sign under a reparameterization. Consider now if det(J) < 0, then the following matrix is obtained

(L˜ ˜g˜ )=J T [( L ˜g )]J = J T [ (L +˜g )]J = J T [(L +˜g )] ij ij ij ij ij ij ij ij and taking the determinant of this matrix gives

det J T [(L +˜g )] = det(J)2 det[(L +˜g )] = 0 ij ij ij ij where the change in sign⇥ changes the sign in⇤ the principal curvatures also.

18Pressley, p. 196

21 The final theorem of this section is stated without proof. However, even though the proof is omitted, the result should be plausible since the principle curvatures are the eigenvalues of the matrix whose components are given by the coecients in (3.16). These are ultimately expressed using the first and second fundamental form, which in the end can be reduced to combinations of the parameterization x and its derivatives, which are smooth.

Theorem 3.7. Let x : U R3 be a surface patch which has no umbilics. Then, the principal curvatures of x are smooth functions! on U 19.

19Pressley, p. 196

22 Chapter 4

Gauss Theorema Egregium

The previous chapter discussed the concept of curvature for a surface. The Gaussian curvature was derived extrinsically as the determinant of the Weingarten map and was then given a very clear geometric meaning, also by extrinsic considerations. In this chapter it will finally be proven that the Gaussian curvature, despite its clearly extrinsic derivation and interpretation, actually is an entirely intrinsic property of a surface. This result is due to Gauss, which found it so extraordinary that he named the result Theorema Egregium, or ”remarkable theorem”. Theorem 4.1 (Theorema Egregium). The Gaussian curvature K of a two-dimensional surface patch x : U R3 depends only on the coecients of the first fundamental form, and is thus an intrinsic quantity! of the surface1. To prove this, however, some more results are first needed.

4.1 The Christo↵el symbols

This section introduces the so called Christo↵el symbols, an important intrinsic quantity of a surface which later will prove very useful.

Theorem 4.2 (Gauss equations). Let x be a surface patch for with gij = xi, xj and Lij = x , x . Then the following holds S h i hh i jii k xij =ijxk + LijN, (4.1)

k 2 where the Christo↵el symbols ij fulfill the following equation 1 @g @g @g k g = il + jl ij . ij kl 2 @uj @ui @ul ✓ ◆ 3 Proof. Since at all points on the surface, the set of vectors xi, N form a basis for R , the second derivatives can be written as linear combinations as { }

k xij = aijxk + bijN,

1K¨uhnel, p. 148 2Pressley, p. 172

23 where ak and b are some unknown coecients. But from equation (3.6), L = x N, and it is ij ij ij ij · clear that bij = Lij. It now remains to determine what the coecients in the xi directions are. Since x N = 0, taking the dot product of x and x gives l · ij l x x = ak g . l · ij ij kl Now, considering the derivatives of the coecients of the first fundamental form @g ij = x x + x x , @ul il · j i · jl @g il = x x + x x , @uj ij · l i · lj @g jl = x x + x x @ui ji · l j · li and using that the derivatives commute, solving for x x results in l · ij 1 @g @g @g x x = il + jl ij = ak g l · ij 2 @uj @ui @ul ij kl ✓ ◆ which completes the proof3. To see that the Christo↵el symbols are intrinsic, writing out the summation index explicitly k gives the following linear system of equations for ij. 1 @g @g @g 1 g +2 g = i1 + j1 ij , ij 11 ij 21 2 @uj @ui @u1 ✓ ◆ (4.2) 1 @g @g @g 1 g +2 g = i2 + j2 ij . ij 12 ij 22 2 @uj @ui @u2 ✓ ◆ The determinant of this system is equal to the determinant of the first fundamental form. The discussion after equation (3.18) shows that the determinant of the first fundamental form is non- k zero. Therefore, this system can be solved for ij using only the coecients of the first fundamental form and their derivatives. However, it is often sucient to simply use the system (4.2) and evaluate it in the particular cases. The main interesting property is that the Christo↵el symbols can be solved in terms of the components of the metric tensor and their derivatives. Therefore, the Christo↵el symbols are intrinsic and any quantity that is expressed using the Christo↵el symbols will also be intrinsic. This will now be used to prove the Theorema Egregium. Before that however, a brief discussion of the notation that will be used is needed4.

4.2 A note on notation

For simplicity, the following comma-notation will be used for derivatives with respect to ul,where the index following the comma shows which parameter to di↵erentiate with respect to. For example, i the derivatives of jk and Lij will be written as @i @L jk =i and ij = L . @ul jk,l @ul ij,l 3do Carmo, p. 154-155 4do Carmo, p. 235-236

24 This notation is used when the quantity being di↵erentiated has indices. However, if, for example, the coecients of the second fundamental form is written out explicitly, e.g. L11 = e and so on, then the comma notation will not follow in the explicit form, thus the derivative of L11 with respect to ui becomes L11,i = ei where the right hand side skips the comma. Another important thing is to remember that summation only occurs when an index is present both as sub- and superscript. Also, when dealing with expressions in index notation, the first- and second fundamental form will be written as gij and Lij. When writing out these expressions explicitly however, it is more convenient to use the notation introduced in Chapter 3; E,F,G and e, f, g for the first- and second fundamental form, respectively. The two ways of expressing the fundamental forms are interchanged somewhat arbitrary, however not in such a way as to cause confusion. The convenience of this notation will hopefully become clear in the following sections.

4.3 Proof of the Theorema Egregium

5 The idea is to use the fact that derivatives commute, i.e. that xikj = xijk . By di↵erentiating equation (4.1), this equality is equivalent to

@ @ l x + L N = l x + L N , (4.3) @uj ik l ik @uk ij l ij which after writing out the derivatives explicitly gives

l l l l ik,j xl +ikxlj + Lik,j N + LikNj =ij,kxl +ijxlk + Lij,kN + LijNk. (4.4)

k By then using the Weingarten equations Ni = ai xk from Section 3.8, the above can be rewritten as l l l l l l ik,j xl +ikxlj + Lik,j N + Likajxl =ij,kxl +ijxlk + Lij,kN + Lijakxl, (4.5) which after collecting terms results in

l l l l l l ik,j + Likaj xl + Lik,j N +ikxlj = ij,k + Lijak xl + Lij,kN +ijxlk. (4.6)

Using equation (4.1) once again gives

l l l r l l l r ik,j + Likaj xl + Lik,j N +ik ljxr + LljN = ij,k + Lijak xl + Lij,kN +ij(lkxr + LlkN) (4.7) l r r l l r r l and since ikljxr =ikrjxl and ijlkxr =ijrkxl, this can be written as

l l r l l l l r l l ik,j + Likaj +ikrj xl + Lik,j +ikLlj N = ij,k + Lijak +ijrk xl + Lij,k +ijLlk N. (4.8) By now equating the coecients for xl and rearranging, this finally results in

l l +r l r l = L al L al . (4.9) ik,j ij,k ik rj ij rk ij k ik j 5Pressley, p. 248

25 The left hand side of equation (4.9) is expressed entirely using the Christo↵el symbols, which by the discussion above are intrinsic. It is therefore possible to define a new intrinsic quantity by Rl =l l +r l r l , (4.10) ijk ik,j ij,k ik rj ij rk which will be called the Riemann curvature tensor. The Riemann curvature tensor gives an inter- esting connection between the intrinsic geometry, captured by the Riemann curvature tensor, and the extrinsic geometry as is clear from equation (4.9). In fact, evaluating equation (4.9) for some specific values on the indices proves the theorem: eF fE fF gE eg f 2 R2 = L a2 L a2 = f e = E = EK, (4.11) 121 12 1 11 2 EG F 2 EG F 2 EG F 2 where the last equality follows from equation (3.18). Equation (4.11) shows that the Gaussian curvature K can be expressed entirely using intrinsic quantities and therefore that K itself is intrinsic. This completes the proof6.

4.4 The Codazzi-Mainardi equations

Equation (4.9) is a non-trivial relation between the first- and second fundamental form that was derived by equating the coecients for the tangent basis vectors in equation (4.8). By instead equating the coecients for the normal vector, another set of non-trivial relations is obtained. These are called the Codazzi-Mainardi equations.

Theorem 4.3 (Codazzi-Mainardi equation). Let x be a surface patch of ,withgij = xi, xj and 7 S h i Lij = xi, xj . Then, hh ii L L +r L r L =0. (4.12) ij,k ik,j ij rk ik rj The rest of this chapter gives some explicit formulas for the Christo↵el symbols, Gaussian curvature and the Codazzi-Mainardi equations when the parameterization is of a particular simple form. These will be needed in the next chapter to prove the final theorem.

4.5 Principal patches

The expressions for the Christo↵el symbols, Gaussian curvature and the Codazzi-Mainardi equations are in general quite complicated. When the parameterization is of the form stated in the next theorem however, these expressions become much simpler. The theorem is stated without a proof. Theorem 4.4. Let p be a point of a surface ,andsupposethatp is not an umbilic. Then, there exists a surface patch x of containing p whichS fulfill S x , x =0, x , x =0. h 1 2i hh 1 2ii A surface patch with these properties is called a principal patch8. In the following subsections, Theorem 4.4 is used for some explicit calculations, beginning with the Christo↵el symbols.

6do Carmo, p. 236-237 7K¨uhnel, p. 147 8Pressley, p. 201

26 4.5.1 The Christo↵el symbols

If g12 = 0, the system of equations (4.2) reduces to 1 @g @g @g 1 g = i1 + j1 ij , ij 11 2 @uj @ui @u1 ✓ ◆ 1 @g @g @g 2 g = i2 + j2 ij ij 22 2 @uj @ui @u2 ✓ ◆ and the Christo↵el symbols are then 1 @g @g @g k = ik + jk ij . (4.13) ij 2g @uj @ui @uk kk ✓ ◆ By using that the Christo↵el symbols and the metric tensor components are symmetric in the lower indices, and that g =0ifi = j, the Christo↵el symbols can be explicitly calculated. ij 6 1 @g @g @g 1 @g E 1 = 11 + 11 11 = 11 = 1 , 11 2g @u1 @u @u1 2g @u1 2E 11 ✓ 1 ◆ 11 1 @g @g @g 1 @g E 1 = 11 + 21 12 = 11 = 2 , 12 2g @u2 @u1 @u1 2g @u2 2E 11 ✓ ◆ 11 1 @g @g @g 1 @g G 1 = 21 + 21 22 = 22 = 1 , 22 2g @u2 @u2 @u1 2g @u1 2E 11 ✓ ◆ 11 (4.14) 1 @g @g @g 1 @g E 2 = 12 + 12 11 = 11 = 2 , 11 2g @u1 @u1 @u2 2g @u2 2G 22 ✓ ◆ 22 1 @g @g @g 1 @g G 2 = 12 + 22 12 = 22 = 1 , 12 2g @u2 @u1 @u2 2g @u1 2G 22 ✓ ◆ 22 1 @g @g @g 1 @g G 2 = 22 + 22 22 = 22 = 2 22 2g @u2 @u2 @u2 2g @u2 2G 22 ✓ ◆ 22 4.5.2 The Gaussian curvature Equations (4.10), (4.11) and (4.14) gives EK = R2 =2 2 +1 2 +2 2 1 2 2 2 = 121 11,2 12,1 11 12 11 22 12 11 12 21 E E G G G2 E G E G E2 G2 = 22 + 2 2 11 + 1 + 1 1 2 2 + 2 1 . 2G 2G2 2G 2G2 4EG 4G2 4EG 4G2

E E G G G2 E G E G E2 G2 2K = 22 2 2 + 11 1 1 1 + 2 2 2 + 1 = EG EG2 EG EG2 2E2G 2EG2 2E2G 2EG2 E E (EG + E G) G G (E G + EG ) = 22 2 2 2 + 11 1 1 1 EG 2E2G2 EG 2E2G2

E E (EG + E G) G G (E G + EG ) 2KpEG = 22 2 2 2 + 11 1 1 1 = pEG 2(EG)3/2 pEG 2(EG)3/2 E G = 2 + 1 . p p ✓ EG◆2 ✓ EG◆1

27 This finally results in the following expression9: 1 @ G @ E K = 1 + 2 . (4.15) p @u1 p @u2 p 2 EG ✓ EG◆ ✓ EG◆ 4.5.3 The Codazzi-Mainardi equations and the principal curvatures The main result of this subsection will be that the Codazzi-Mainardi equations yield a convenient expression for the derivatives of the principle curvatures and the coecients of the first fundamental form when the parameterization is a principle patch10. From equation (4.12), the Codazzi-Mainardi equations are: L L +r L r L =0. ij,k ik,j ij rk ik rj By chosing the indices to be i =1,j =1,k = 2 and using that L12 = L21 = 0, this reduces to L +2 L 1 L =0. 11,2 11 22 12 11 Using (4.14) for the Christo↵el symbols and changing to the explicit notation for the coecients of the first- and second fundamental form, the above can be written as 1 g e e = E + . (4.16) 2 2 2 G E ⇣ ⌘ The same calculations but for i =2,j =2,k =1resultsin 1 g e g = G + . (4.17) 1 2 1 G E ⇣ ⌘ Now, since the parameterization is a principal patch, (3.16) shows that the matrix of the Weingarten map is e/E 0 0 g/G ✓ ◆ and the principle curvatures are given by

1 = e/E, 2 = g/G. (4.18) Using (4.16), (4.17) and (4.18) then results in E G  = 2 (  ), = 1 (  ). (4.19) 1,2 2E 2 1 2,1 2G 1 2 From (4.19), the following expressions for the second derivatives of the coecients of the first fundamental are obtained: @ 2E 2(E  + E )   E = 1,2 = 2 1,2 1,22 2E 2,2 1,2 , 22 @u2     1,2 (  )2 ✓ 2 1 ◆ 2 1 2 1 (4.20)

@ 2G2,1 2(G12,1 + G2,11) 1,1 2,1 G11 = 1 = 2G2,1 2 , @u 1 2 1 2 (  ) ✓ ◆ 1 2 which holds when the point of evaluation is not an umbilic, i.e. when 1 = 2.Thesewillbeused in the proof of an important lemma in the next chapter. 6 9Pressley, p. 252-253 10Pressley, p. 251

28 Chapter 5

The Final Result

5.1 Statement of final result

The final result for this project can now be stated, which this whole chapter is dedicated to proving. The theorem concerns the global structure of a surface, which has been of interest for some time as this result was known since 18991. What makes the global structure interesting is that so far, information about surfaces have only been described locally. For example, Theorem 3.4 states that if all the points of a surface are umbilical with non-zero principal curvature, then the surface is part of an open subset of a sphere. By imposing that the Gaussian curvature is constant for the entire surface then conclusions can be drawn about the surface globally, which is what is to be proven. Theorem 5.1. If is a connected and compact surface with constant Gaussian curvature, then is a sphere2. S S

Note that this theorem does not depend on the surface only containing umbilical points with non- zero principal curvatures, that comes later. Several lemmas will be needed to prove this theorem. To begin with, an additional piece of information can be stated about compact surfaces and its Gaussian curvature. Lemma 5.1. If is a compact surface, then there exists a point on the surface such that the Gaussian curvatureS is positive there3. Proof. (Lemma 5.1) Take a curve (t) with unit speed lying on the surface such that the curve passes through a point p at t = t0. Choose a surface patch x with p in the image of this surface patch. Since the surface is assumed to be compact, and the norm function, : R3 R,isa continuous function then there exists a point on the surface such that the norm reachesk·k a! maximum at a point p.Thus; (t) has a maximum at t . Di↵erentiating results in k k 0 d (t) =0 = (t ) ˙ (t )=0. dtk k ) 0 · 0 t=t0 1 K¨uhnel, p. 189 2Pressley, p. 261 3Pressley, p. 212

29 This equation shows that (t0) is perpendicular to the tangent plane Tp . The surface patch was assumed to contain p in its image, thus a normal vector to the surface atS point p can be chosen as p N = . ± p k k This was yielded from considering the first derivative evaluated at t = t0. Next, consider the second derivative of (t) at t = t : k k 0 d2 (t) 0= (t ) ¨(t )+1 0. dt2 k k  ) 0 · 0  t=t0 The normal curvature,  = ¨ N (recall from chapter 3), at t then becomes  = ¨(t ) N. n · 0 n 0 · However, p was defined as (t0), such that N = (t0)/ p . With this, the normal curvature becomes ± k k ¨(t ) (t ) 1 1  = 0 · 0 =  or  n ± p ) n p n  p k k k k k k where the inequality that arises from the second derivative has been used. The principal curvatures has the important property that they are the maximum and minimum of the normal curvature to a surface at a point on the surface. Thus, the principal curvatures are either both less than or equal to 1/ p or both larger than or equal to 1/ p . Either case, the Gaussian curvature results in k k k k K 1/ p 2 which is positive4. k k The next lemma needed concerns surface patches that contain a point that is not an umbilic point, which states that the Gaussian curvature will be non-positive there.

Lemma 5.2. Let x : U R3 be a surface patch with p in its image such that p is not a umbilic point. Assume further that! the principal curvatures,  and  ,ofx are such that   and that 1 2 1 2 1 has a local maximum at p and 2 has a local minimum at p. Then the Gaussian curvature of x at p is less than or equal to zero5. Proof. (Lemma 5.2) According to theorem 4.4, where the condition that p is not an umbilic is satisfied, the point p can be covered by a principal patch. Furthermore, since p is a stationary point for both 1 and 2 then E2 = G1 = 0 (follows from equation (4.12)). This results in the Gaussian curvature being reduced to 1 K = (G + E ) 2EG 11 22 which follows from equation (4.15). From equation (4.20), and again using the fact that E2 = G1 =0 and that the principal curvatures are stationary at p,then 2E 2G E = 1,22 ,G= 2,11 22   11   1 2 1 2 which in turn enables the Gaussian curvature to be written as 1 E G K = 1,22 2,11 . EG   ✓ 1 2 ◆ 4Pressley, p. 212-213 5Pressley, p. 261

30 It is known that E = x x , G = x x , and   are all positive, and that 1 · 1 2 · 2 1 2  0, 0 1,22  2,11 where the latter stems from the principal curvatures being local maximum and minimum, respec- tively, at p. Hence, the latest expression for the Gaussian curvature shows that K 06.  All the necessary information, theorems, and lemmas are in place to give a proof of Theorem 5.1.

5.2 Proof of the theorem

The idea of the proof is to show that each point on the surface, call it , is an umbilic. Then, by Theorem 3.4, is an open subset of a sphere. But since is compactS it is also closed7.A well known theoremS from topology, here taken for granted, statesS that the only two subsets of a connected set that are simultaneously open and closed is the empty set and the set itself. Since by assumption is non-empty and connected (all surfaces in this project are assumed connected). ThisS shows that is in fact the entire sphere. The only thing to prove then is that each point on is an umbilic. S S The first thing to note is that, by Lemma 5.1 and the assumption that K is constant, the Gaussian curvature is strictly positive. Next, consider the function

J =(  )2 1 2 defined on ,wherei are the principal curvatures. This function does not take the labeling of the principal curvaturesS into account, and by Lemma 3.6 the principal curvatures either stay the same sign or both change sign during reparameterizations, so J is well defined. What needs to be proven then is that J = 0 for all point on , and this will be done by a contradiction. Assume that there exists someS point on where J = 0. Since the eigenvalues of a linear operator are continuously dependent on theS linear operator6 8 and since the Weingarten map is defined everywhere on the compact surface then J is continuous everywhere. Hence, J attains its maximum at some point p . Also, byS Lemma 3.6 and the fact   = K>0 by choosing 2S 1 2 a suitable surface patch x : U R3, both principal curvatures can be assumed to be positive and ! it can also be assumed that 1 >2 at p. Again using the continuity of the principle curvatures, these conditions can be assumed to hold at all points in U. This can always be achieved by, if necessary, choosing a smaller subset of U where these conditions hold. Since p was chosen to be the point where J has its maximum, J must decrease at all points around p.ButJ is increasing with 1. To see this, note that K = 12 is constant so 2 = K/1 and J can be rewritten as K 2 J =  1  ✓ 1 ◆ which, since 1 > K/1 clearly increases if 1 increases. For J to have a maximum at p then, 1 must decrease at all points around p and thus have a local maximum at p.Then,sinceK is

6Pressley, p. 261-262 7Rudin, p. 37 8Serre, p. 88-89

31 constant and K = 12, this also implies that 2 has a local minimum at p and by Lemma 5.2 that K 0. This contradicts the assumption that K>0, which in turn shows that J = 0, i.e. that  =  for all points on . This completes the proof9. 1 2 S

9Pressley, p. 262

32 Chapter 6

End note

Throughout this project, a restriction has been made to look at surfaces embedded in three- dimensional Euclidean space. While several important properties were studied in this setting, it is worth noting that in many cases there is a natural generalization to higher-dimensional embed- dings, and in some cases no embedding at all is taken into account. One such case is Riemannian geometry. The step one needs to take from extrinsic and intrinsic geometry of surfaces to Rieman- nian geometry involve some similar concepts and notions mentioned above. One such notion is the first fundamental form, which was shown to be an intrinsic quantity. The second notion generalizes the ”surface” used in this project into a global structure called a manifold. No technical details will be stated. An intrinsic geometry such as the Riemannian geometry is very successful at describing the general theory of relativity. The success is due to the fact that spacetime is not embedded in any ”larger” space, thus only intrinsic properties about spacetime can be known. Even more of interest is that the central result of this project stated that the shape of a surface depended on an intrinsic quantity, and similar statements can be made about the shape of spacetime. All in all, the goal of the project was to develop enough theory about surfaces embedded in three-dimensional Euclidian space in order to prove the theorem stated in the introduction. In order to reach this goal, a formal definition of surfaces in R3 was given. Following this, several properties about surfaces were stated, such as the tangent plane of a surface at a point which also gives rise to the unit normal vector of a surface at the same point. From there on, the extrinsic and intrinsic properties of surfaces were studied, where both the first and second fundamental form was formulated, and the Gaussian curvature was derived extrinsically. An important result within this context was that a surface is an open subset of a sphere if every point on the surface is an umbilic and had non-zero principal curvature. Following this, Gauss Theorema Egregium showed that the Gaussian curvature is in fact an intrinsic quantity despite its extrinsic derivation. Finally, all necessary theory was needed to prove the original statement which concluded this project.

33 Bibliography

[1] Pressley, Andrew. (2012). Elementary Di↵erential Geometry. 2nd ed. London: Springer. [2] K¨uhnel, Wolfgang. (2015). Di↵erential Geometry; Curves-Surfaces-Manifolds 3rd ed. Prov- idence: American Mathematical Society. [3] do Carmo, Manfredo P. (1976). Di↵erential Geometry of Curves and Surfaces. New Jersey: Prentice-Hall, inc. [4] Vas, Lia. Curves.[pdf].https://liavas.net/courses/math430/files/Curves.pdf (Accessed 23/04/19) [5] Vas, Lia. Measuring Lengths - The First Fundamental Form.[pdf].https://liavas.net/ courses/math430/files/Surfaces_part2.pdf (Accessed 23/04/19) [6] Gluck, Herman. 3. The Geometry of the Gauss Map.[pdf].https://www.math.upenn.edu/ ~shiydong/Math501X-3-GaussMap.pdf (Accessed 23/04/19) [7] Alexandrov, A.D. (2006). Part II, Intrinsic Geometry of Convex Surfaces. Boca Raton: Chapman & Hall/CRC Taylor & Francis Group. [8] Rudin, Walter. (1976). Principles of Mathematical Analysis 3rd ed. McGraw Hill.

[9] Serre, Denis. (2010). Matrices Theory and Applications. 2nd ed. New York: Springer.

34 www.kth.se www.kth.se