DOCSLIB.ORG

Classical Differential Geometry of Two-Dimensional Surfaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Classical Differential Geometry of Two-Dimensional Surfaces Classical di®erential geometry of two-dimensional surfaces 1 Basic de¯nitions This section gives an overview of the basic notions of di®erential geometry for two- dimensional surfaces. It follows mainly Kreyszig [Kre91] in its discussion. De¯nition of a surface Let us consider the vector function X(»1; »2) R3 with 2 X : R2 ¥ (»1; »2) X(»1; »2) U R3 ; (1) 3 7! 2 ½ where ¥ is an open subset of R2. Let X(»1; »2) be of class r 1 in ¥, which means ¸ that one of its component functions Xi (i x; y; z ) is of class r and the other 1 2 f g @(Xx;Xy;Xz) ones are at least of this class. Let furthermore the Jacobian matrix @(»1;»2) be of rank 2 in ¥ which implies that the vectors @X ea := = @aX ; a 1; 2 ; (2) @»a 2 f g are linearly independent. The mapping (1) then de¯nes a smooth two-dimensional surface patch U embedded in three-dimensional Euclidean space R3 with coordi- nates »1 and »2 (see Fig. 1). A union § of surface patches is called a surface if two arbitrary patches U and U 0 of § can be joined by ¯nitely many patches 0 U = U1; U2; : : : ; Un¡1; Un = U in such a way that the intersection of two subsequent patches is again a surface patch [Kre91, p. 76]. To simplify the following let us restrict ourselves to a surface that can be covered by one patch U only. The vectors ea, de¯ned in Eqn. (2), are the tangent vectors of the surface. They are not normalized in general. Together with the unit normal e e n := 1 £ 2 ; (3) e e j 1 £ 2j they form a local basis (local frame) in R3 (see Fig. 2): ea n = 0 ; and n n = 1: (4) ¢ ¢ 1 A function of one or several variables is called a function of class r if it possesses continous partial derivatives up to order r. 1 z Σ U y X ξ 2 Ξ x ξ1 Figure 1: Parametrization of a surface n e2 γ e1 z X y x Figure 2: Local frame on the surface 2 The metric tensor (¯rst fundamental form) With the tangent vectors ea, one can de¯ne the metric tensor (also called the ¯rst fundamental form) gab := ea eb : (5) ¢ This covariant second rank tensor is symmetric (gab = gba) and positive de¯nite [Kre91, p. 86]. It helps to determine the in¯nitesimal Euclidean distance in terms of the coordinate di®erentials [Kre91, p. 82] 2 1 1 2 2 1 2 2 1 2 2 ds = [X(» + d» ; » + d» ) X(» ; » )] = (e1 d» + e2 d» ) a 2 ¡a b = (ea d» ) = (ea eb) d» d» a b ¢ = gab d» d» ; (6) where the sum convention is used in the last two lines (see App. ??). The con- travariant dual tensor of the metric may be de¯ned via cb b 1; if a = b gac g := ± := ; (7) a 0; if a = b ( 6 b where ±a is the Kronecker symbol. The metric and its inverse can be used to raise and lower indices in tensor equations. Consider for instance the second rank tensor tab: cb b c Raising: tac g = ta ; and lowering: ta gcb = tab : (8) The determinant of the metric2 g := det g = gab = g g g g (9) j j 11 22 ¡ 12 21 can be exploited to calculate the in¯nitesimal area element dA: let be the angle between e1 and e2 (see Fig. 2). Then e e 2 = e 2 e 2 sin2 = g g (1 cos2 ) = g g (e e )2 j 1 £ 2j j 1j j 2j 11 22 ¡ 11 22 ¡ 1 ¢ 2 = g g g g = g ; (10) 11 22 ¡ 12 12 and thus dA = e e d»1 d»2 = pg d2» : (11) j 1 £ 2j The covariant derivative The partial derivative @a is itself not a tensor. One therefore de¯nes the covariant a1a2:::an derivative a on a tensor t r b1b2:::bm a1a2:::an a1a2:::an ct = @ct r b1b2:::bm b1b2:::bm d a2:::an a1 a1d:::an a2 a1a2:::d an + tb1b2:::bm ¡dc + tb1b2:::bm ¡dc + : : : + tb1b2:::bm ¡dc ta1a2:::an ¡ d ta1a2:::an ¡ d : : : ta1a2:::an ¡ d ; (12) ¡ d b2:::bm b1c ¡ b1d:::bm b2c ¡ ¡ b1b2:::d bmc 2 Note that g is the matrix consisting of the metric tensor components gab. 3 c where the ¡ab are the Christo®el symbols of the second kind with c c ¡ = (@aeb) e ; (13) ab ¢ and a is now a tensor. For the covariant di®erentiation of sums and products of tensorsr the usual rules of di®erential calculus hold. The metric-compatible Laplacian a ¢ can be de¯ned as ¢ := a . r r Note in particular that c aeb = @aeb ¡ab ec ; and (14) r bc¡ agbc = ag = ag = 0 : (15) r r r Equation (15) is also called the Lemma of Ricci. It implies that raising and lowering of indices commutes with the process of covariant di®erentiation. Orientable surfaces The orientation of the normal vector n in one point S of the surface depends on the choice of the coordinate system [Kre91, p. 108]: exchanging, for instance, »1 and »2 also ips n by 180 degrees. A surface is called orientable if no closed curve through any point S of the surface exists which causes the sense of n to change whenC displacing n continuously from S along back to S. An example of a surface that is not orientable is the MÄobius strip. C The extrinsic curvature tensor (second fundamental form) Two surfaces may have the same metric tensor gab but di®erent curvature properties in R3. In order to describe such properties let us consider a surface § of class3 r 2 and a curve of the same class on § with the parametrization X(»1(s); »2(s))¸on §, where s isCthe arc length of the curve (see Fig. 3). At every point of the curve where its curvature k > 0, one may de¯ne a moving trihedron t; p; b where t = X_ is the unit tangent vector, p = t_= t_ = t_=k is the unit principalf normalg vector, and b = t p is the unit binormal vectorj jof the curve.4 Furthermore, let ´ be the angle between£the unit normal vector n of the surface and the unit principal normal vector p of the curve with cos ´ = p n (see again Fig. 3). The curvature k of the curve can then be decomposed into a¢part which is due to the fact that the surface is curved in R3 and a part due to the fact that the curve itself is curved. The former will be called the normal curvature Kn, the latter the geodesic curvature Kg. One de¯nes: K := t_ n = k (p n) = k cos ´ ; and (16) n ¡ ¢ ¡ ¢ ¡ K := t (t_ n) = k t (p n) = k sin ´ sign (n b) : (17) g ¢ £ ¢ £ ¢ 3 This means that its parametrization X(»1; »2) is of class r 2. 4 The dot denotes the derivative with respect to the arc length¸ s. 4 Σ n η S t p C b Figure 3: Curve on a surface Here, we are interested in the curvature properties of the surface. Therefore, the 5 normal curvature Kn is the relevant quantity that has to be studied a bit further. The vector t_ may be written as d t_ = XÄ = (e »_a) = (@ e ) »_a»_b + e »Äa : (18) ds a b a a Thus, Eqn. (16) turns into a b K = k cos ´ = ( n @aeb) »_ »_ ; (19) n ¡ ¡ ¢ where it has been exploited that @aeb = @bea. The expression in brackets is the extrinsic curvature tensor or second fundamental form Kab := n @aeb = ea @bn : (20) ¡ ¢ ¢ It is a symmetric covariant second rank tensor such as the metric. The second relation in Eqn. (20) follows if one di®erentiates the ¯rst equation of (4) with respect to »a. The extrinsic curvature can be written covariantly: Kab := n aeb : (21) ¡ ¢ r This is possible because @aeb di®ers from aeb only by terms proportional to the r tangent vectors ec, which vanish when multiplied by n (see Eqn. (14)). 5 The minus sign in the de¯nition of Kn, Eqn. (16), is unfortunately a matter of convention and is here chosen in accordance to the literature where the surface stress tensor for uid membranes has been introduced [CG02, Guv04]. A sphere with outward pointing unit normal has a positive normal curvature then. Note that this di®ers from Ref. [Kre91]. 5 One can easily see from Eqn. (20) that Kab has got something to do with curvature: at every point of the surface it measures the change of the normal vector in R3 for an in¯nitesimal displacement in the direction of a coordinate curve. To learn more about the normal curvature let us consider a reparametrization of the curve with the new parameter t. One gets C d»a dt »a0 »_a = = ; (22) dt ds s0 where 0 denotes the derivative with respect to t. Equation (19) thus takes the form a0 b0 a0 b0 a b K » » (6) K » » K d» d» K = K »_a»_b = ab = ab = ab : (23) n ab 0 2 a0 b0 a b (s ) gab » » gab d» d» For a ¯xed point S, Kab and gab are ¯xed as well. The value of Kn then only depends on the direction of the tangent vector t of the curve. One may search for extremal values of Kn at S by rewriting Eqn.
Load more

Recommended publications
  • Geometry of Curves
    Appendix A Geometry of Curves “Arc, amplitude, and curvature sustain a similar relation to each other as time, motion and velocity, or as volume, mass and density.” Carl Friedrich Gauss The rest of this lecture notes is about geometry of curves and surfaces in R2 and R3. It will not be covered during lectures in MATH 4033 and is not essential to the course. However, it is recommended for readers who want to acquire workable knowledge on Differential Geometry. A.1. Curvature and Torsion A.1.1. Regular Curves. A curve in the Euclidean space Rn is regarded as a function r(t) from an interval I to Rn. The interval I can be finite, infinite, open, closed n n or half-open. Denote the coordinates of R by (x1, x2, ... , xn), then a curve r(t) in R can be written in coordinate form as: r(t) = (x1(t), x2(t),..., xn(t)). One easy way to make sense of a curve is to regard it as the trajectory of a particle. At any time t, the functions x1(t), x2(t), ... , xn(t) give the coordinates of the particle in n R . Assuming all xi(t), where 1 ≤ i ≤ n, are at least twice differentiable, then the first derivative r0(t) represents the velocity of the particle, its magnitude jr0(t)j is the speed of the particle, and the second derivative r00(t) represents the acceleration of the particle. As a course on Differential Manifolds/Geometry, we will mostly study those curves which are infinitely differentiable (i.e.
    [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]
  • Basics of the Differential Geometry of Surfaces
    Chapter 20 Basics of the Differential Geometry of Surfaces 20.1 Introduction The purpose of this chapter is to introduce the reader to some elementary concepts of the differential geometry of surfaces. Our goal is rather modest: We simply want to introduce the concepts needed to understand the notion of Gaussian curvature, mean curvature, principal curvatures, and geodesic lines. Almost all of the material presented in this chapter is based on lectures given by Eugenio Calabi in an upper undergraduate differential geometry course offered in the fall of 1994. Most of the topics covered in this course have been included, except a presentation of the global Gauss–Bonnet–Hopf theorem, some material on special coordinate systems, and Hilbert’s theorem on surfaces of constant negative curvature. What is a surface? A precise answer cannot really be given without introducing the concept of a manifold. An informal answer is to say that a surface is a set of points in R3 such that for every point p on the surface there is a small (perhaps very small) neighborhood U of p that is continuously deformable into a little flat open disk. Thus, a surface should really have some topology. Also,locally,unlessthe point p is “singular,” the surface looks like a plane. Properties of surfaces can be classified into local properties and global prop- erties.Intheolderliterature,thestudyoflocalpropertieswascalled geometry in the small,andthestudyofglobalpropertieswascalledgeometry in the large.Lo- cal properties are the properties that hold in a small neighborhood of a point on a surface. Curvature is a local property. Local properties canbestudiedmoreconve- niently by assuming that the surface is parametrized locally.
    [Show full text]
  • Some Examples of Critical Points for the Total Mean Curvature Functional
    Proceedings of the Edinburgh Mathematical Society (2000) 43, 587-603 © SOME EXAMPLES OF CRITICAL POINTS FOR THE TOTAL MEAN CURVATURE FUNCTIONAL JOSU ARROYO1, MANUEL BARROS2 AND OSCAR J. GARAY1 1 Departamento de Matemdticas, Universidad del Pais Vasco/Euskal Herriko Unibertsitatea, Aptdo 644 > 48080 Bilbao, Spain ([email protected]; [email protected]) 2 Departamento de Geometria y Topologia, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain ([email protected]) (Received 31 March 1999) Abstract We study the following problem: existence and classification of closed curves which are critical points for the total curvature functional, defined on spaces of curves in a Riemannian manifold. This problem is completely solved in a real space form. Next, we give examples of critical points for this functional in a class of metrics with constant scalar curvature on the three sphere. Also, we obtain a rational one-parameter family of closed helices which are critical points for that functional in CP2(4) when it is endowed with its usual Kaehlerian structure. Finally, we use the principle of symmetric criticality to get equivariant submanifolds, constructed on the above curves, which are critical points for the total mean curvature functional. Keywords: total curvature; critical points; total tension AMS 1991 Mathematics subject classification: Primary 53C40; 53A05 1. Introduction The tension field of a submanifold is the Euler-Lagrange operator associated with the energy of the submanifold [9] and it is precisely the mean curvature vector field. Thus, the amount of tension that a submanifold receives at one point is measured by the mean curvature function at that point, a.
    [Show full text]
  • Arxiv:1709.02078V1 [Math.DG]
    DENSITY OF A MINIMAL SUBMANIFOLD AND TOTAL CURVATURE OF ITS BOUNDARY Jaigyoung Choe∗ and Robert Gulliver Abstract Given a piecewise smooth submanifold Γn−1 ⊂ Rm and p ∈ Rm, we define the vision angle Πp(Γ) to be the (n − 1)-dimensional volume of the radial projection of Γ to the unit sphere centered at p. If p is a point on a stationary n-rectifiable set Σ ⊂ Rm with boundary Γ, then we show the density of Σ at p is ≤ the density at its vertex p of the n−1 cone over Γ. It follows that if Πp(Γ) is less than twice the volume of S , for all p ∈ Γ, then Σ is an embedded submanifold. As a consequence, we prove that given two n-planes n n Rm n R1 , R2 in and two compact convex hypersurfaces Γi of Ri ,i =1, 2, a nonflat minimal submanifold spanned by Γ := Γ1 ∪ Γ2 is embedded. 1 Introduction Fenchel [F1] showed that the total curvature of a closed space curve γ ⊂ Rm is at least 2π, and it equals 2π if and only if γ is a plane convex curve. F´ary [Fa] and Milnor [M] independently proved that a simple knotted regular curve has total curvature larger than 4π. These two results indicate that a Jordan curve which is curved at most double the minimum is isotopically simple. But in fact minimal surfaces spanning such Jordan curves must be simple as well. Indeed, Nitsche [N] showed that an analytic Jordan curve in R3 with total curvature at most 4π bounds exactly one minimal disk.
    [Show full text]
  • AN INTRODUCTION to the CURVATURE of SURFACES by PHILIP ANTHONY BARILE a Thesis Submitted to the Graduate School-Camden Rutgers
    AN INTRODUCTION TO THE CURVATURE OF SURFACES By PHILIP ANTHONY BARILE A thesis submitted to the Graduate School-Camden Rutgers, The State University Of New Jersey in partial fulfillment of the requirements for the degree of Master of Science Graduate Program in Mathematics written under the direction of Haydee Herrera and approved by Camden, NJ January 2009 ABSTRACT OF THE THESIS An Introduction to the Curvature of Surfaces by PHILIP ANTHONY BARILE Thesis Director: Haydee Herrera Curvature is fundamental to the study of differential geometry. It describes different geometrical and topological properties of a surface in R3. Two types of curvature are discussed in this paper: intrinsic and extrinsic. Numerous examples are given which motivate definitions, properties and theorems concerning curvature. ii 1 1 Introduction For surfaces in R3, there are several different ways to measure curvature. Some curvature, like normal curvature, has the property such that it depends on how we embed the surface in R3. Normal curvature is extrinsic; that is, it could not be measured by being on the surface. On the other hand, another measurement of curvature, namely Gauss curvature, does not depend on how we embed the surface in R3. Gauss curvature is intrinsic; that is, it can be measured from on the surface. In order to engage in a discussion about curvature of surfaces, we must introduce some important concepts such as regular surfaces, the tangent plane, the first and second fundamental form, and the Gauss Map. Sections 2,3 and 4 introduce these preliminaries, however, their importance should not be understated as they lay the groundwork for more subtle and advanced topics in differential geometry.
    [Show full text]
  • Curvatures of Smooth and Discrete Surfaces
    Curvatures of Smooth and Discrete Surfaces John M. Sullivan Abstract. We discuss notions of Gauss curvature and mean curvature for polyhedral surfaces. The discretizations are guided by the principle of preserving integral relations for curvatures, like the Gauss/Bonnet theorem and the mean-curvature force balance equation. Keywords. Discrete Gauss curvature, discrete mean curvature, integral curvature rela- tions. The curvatures of a smooth surface are local measures of its shape. Here we con- sider analogous quantities for discrete surfaces, meaning triangulated polyhedral surfaces. Often the most useful analogs are those which preserve integral relations for curvature, like the Gauss/Bonnet theorem or the force balance equation for mean curvature. For sim- plicity, we usually restrict our attention to surfaces in euclidean three-space E3, although some of the results generalize to other ambient manifolds of arbitrary dimension. This article is intended as background for some of the related contributions to this volume. Much of the material here is not new; some is even quite old. Although some references are given, no attempt has been made to give a comprehensive bibliography or a full picture of the historical development of the ideas. 1. Smooth curves, framings and integral curvature relations A companion article [Sul08] in this volume investigates curves of finite total curvature. This class includes both smooth and polygonal curves, and allows a unified treatment of curvature. Here we briefly review the theory of smooth curves from the point of view we arXiv:0710.4497v1 [math.DG] 24 Oct 2007 will later adopt for surfaces. The curvatures of a smooth curve γ (which we usually assume is parametrized by its arclength s) are the local properties of its shape, invariant under euclidean motions.
    [Show full text]
  • Shapewright: Finite Element Based Free-Form Shape Design
    ShapeWright: Finite Element Based Free-Form Shape Design by George Celniker S.M., Mechanical Engineering Massachusetts Institute of Technology January, 1981 B.S., Mechanical Engineering Massachusetts Institute of Technology May, 1979 Submitted to the Department of Mechanical Engineering in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mechanical Engineering at the Massachusetts Institute of Technology September, 1990 © Massachusetts Institute of Technology, 1990, All rights reserved. Signature of Author . ... , , , Department of Mechariical Engineering August 3, 1990 Certified by Professor David C. Gossard Thesis Supervisor Accepted by Ain A. Sonin Chairman, Department Committee MASSACHUITTS INSTITOiI OF TECHIvn .:.. NOV 0 8 1990 LIBRARIEb ShapeWright: Finite Element Based Free-Form Shape Design by George Celniker Submitted to the Department of Mechanical Engineering on September, 1990 inpartial fulfillment of the requirements for the degree of Doctor of Philosophy inMechanical Engineering. Abstract The mathematical foundations and an implementation of a new free-form shape design paradigm are developed. The finite element method is applied to generate shapes that minimize a surface energy function subject to user-specified geometric constraints while responding to user-specified forces. Because of the energy minimization these surfaces opportunistically seek globally fair shapes. It is proposed that shape fairness is a global and not a local property. Very fair shapes with C1 continuity are designed. Two finite elements, a curve segment and a triangular surface element, are developed and used as the geometric primitives of the approach. The curve segments yield piecewise C1 Hermite cubic polynomials. The surface element edges have cubic variations in position and parabolic variations in surface normal and can be combined to yield piecewise C1 surfaces.
    [Show full text]
  • Combinatorial, Bakry-Émery, Ollivier's Ricci Curvature Notions and Their
    Combinatorial, Bakry-Émery, Ollivier’s Ricci curvature notions and their motivation from Riemannian geometry Supanat Kamtue arXiv:1803.08898v1 [math.CO] 23 Mar 2018 Department of Mathematical Sciences Durham University, UK Abstract In this survey, we study three different notions of curvature that are defined on graphs, namely, combinatorial curvature, Bakry-Émery curvature, and Ollivier’s Ricci curvature. For each curvature notion, the definition and its motivation from Riemannian geometry will be explained. Moreover, we bring together some global results and geometric concepts in Riemannian geometry that are related to curvature (e.g. Bonnet-Myers theorem, Laplacian opera- tor, Lichnerowicz theorem, Cheeger constant), and then compare them to the discrete analogues in some (if not all) of the discrete curvature notions. The structure of this survey is as follows: the first chapter is dedicated to relevant background in Riemannian geometry. Each following chapter is focussing on one of the discrete curvature notions. This survay is an MSc dissertation in Mathematical Sciences at Durham University. Contents 1 Background in Riemannian Geometry3 1.1 Gauss-Bonnet, Cartan-Hadamard, and Cheeger constant . .3 1.2 Laplacian operator and Bochner’s formula . .8 1.3 Eigenvalues of Laplacian and Lichnerowicz theorem . 14 1.4 Bonnet-Myers theorem . 15 1.5 Average distance between two balls . 17 1.6 Examples of graphs and manifolds . 21 2 Combinatorial Curvature 24 2.1 Planar tessellations . 24 2.2 Combinatorial Gauss-Bonnet and Cartan-Hadamard . 27 2.3 Cheeger constant and isoperimetric inequality on graphs . 29 2.4 Combinatorial Bonnet-Myers . 32 3 Bakry-Émery Curvature 35 3.1 CD inequality and Γ-calculus .
    [Show full text]
  • Fundamental Forms of Surfaces and the Gauss-Bonnet Theorem
    FUNDAMENTAL FORMS OF SURFACES AND THE GAUSS-BONNET THEOREM HUNTER S. CHASE Abstract. We define the first and second fundamental forms of surfaces, ex- ploring their properties as they relate to measuring arc lengths and areas, identifying isometric surfaces, and finding extrema. These forms can be used to define the Gaussian curvature, which is, unlike the first and second fun- damental forms, independent of the parametrization of the surface. Gauss's Egregious Theorem reveals more about the Gaussian curvature, that it de- pends only on the first fundamental form and thus identifies isometries be- tween surfaces. Finally, we present the Gauss-Bonnet Theorem, a remarkable theorem that, in its global form, relates the Gaussian curvature of the surface, a geometric property, to the Euler characteristic of the surface, a topological property. Contents 1. Surfaces and the First Fundamental Form 1 2. The Second Fundamental Form 5 3. Curvature 7 4. The Gauss-Bonnet Theorem 8 Acknowledgments 12 References 12 1. Surfaces and the First Fundamental Form We begin our study by examining two properties of surfaces in R3, called the first and second fundamental forms. First, however, we must understand what we are dealing with when we talk about a surface. Definition 1.1. A smooth surface in R3 is a subset X ⊂ R3 such that each point has a neighborhood U ⊂ X and a map r : V ! R3 from an open set V ⊂ R2 such that • r : V ! U is a homeomorphism. This means that r is a bijection that continously maps V into U, and that the inverse function r−1 exists and is continuous.
    [Show full text]
  • Geometry of Surfaces 1 4.1
    M435: INTRODUCTION TO DIFFERENTIAL GEOMETRY MARK POWELL Contents 4. Geometry of Surfaces 1 4.1. Smooth patches 1 4.2. Tangent spaces 3 4.3. Vector fields and covariant derivatives 4 4.4. The normal vector and the Gauss map 7 4.5. Change of parametrisation 7 4.6. Linear Algebra review 14 4.7. First Fundamental Form 14 4.8. Angles 16 4.9. Areas 17 4.10. Effect of change of coordinates on first fundamental form 17 4.11. Isometries and local isometries 18 References 21 4. Geometry of Surfaces 4.1. Smooth patches. Definition 4.1. A smooth patch of a surface in R3 consists of an open subset U ⊆ R2 and a smooth map r: U ! R3 which is a homeomorphism onto its image r(U), and which has injective derivative. The image r(U) is a smooth surface in R3. Recall that the injective derivative condition can be checked by computing ru ^ rv =6 0 everywhere. A patch gives us coordinates (u; v) on our surface. The curves α(t) = r(t; v) and β(t) = r(u; t), for fixed u; v, are the coordinate lines on our surface. The lines of constant longitude and latitude on the surface of the earth are examples. Smooth means that all derivatives exist. In practice, derivatives up to third or fourth order at most are usually sufficient. Example 4.2. The requirement that r be injective means that we sometimes have to restrict our domain, and we may not cover the whole of some compact surface with one patch.
    [Show full text]
  • The Second Fundamental Form. Geodesics. the Curvature Tensor
    Differential Geometry Lia Vas The Second Fundamental Form. Geodesics. The Curvature Tensor. The Fundamental Theorem of Surfaces. Manifolds The Second Fundamental Form and the Christoffel symbols. Consider a surface x = @x @x x(u; v): Following the reasoning that x1 and x2 denote the derivatives @u and @v respectively, we denote the second derivatives @2x @2x @2x @2x @u2 by x11; @v@u by x12; @u@v by x21; and @v2 by x22: The terms xij; i; j = 1; 2 can be represented as a linear combination of tangential and normal component. Each of the vectors xij can be repre- sented as a combination of the tangent component (which itself is a combination of vectors x1 and x2) and the normal component (which is a multi- 1 2 ple of the unit normal vector n). Let Γij and Γij denote the coefficients of the tangent component and Lij denote the coefficient with n of vector xij: Thus, 1 2 P k xij = Γijx1 + Γijx2 + Lijn = k Γijxk + Lijn: The formula above is called the Gauss formula. k The coefficients Γij where i; j; k = 1; 2 are called Christoffel symbols and the coefficients Lij; i; j = 1; 2 are called the coefficients of the second fundamental form. Einstein notation and tensors. The term \Einstein notation" refers to the certain summation convention that appears often in differential geometry and its many applications. Consider a formula can be written in terms of a sum over an index that appears in subscript of one and superscript of P k the other variable. For example, xij = k Γijxk + Lijn: In cases like this the summation symbol is omitted.
    [Show full text]