DOCSLIB.ORG

Elementary Differential Geometry and the Gauss-Bonnet Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Elementary Differential Geometry and the Gauss-Bonnet Theorem ELEMENTARY DIFFERENTIAL GEOMETRY AND THE GAUSS-BONNET THEOREM DUSTIN BURDA Abstract. In this paper we discuss examples of the classical Gauss-Bonnet theorem under constant positive Gaussian curvature and zero Gaussian cur- vature. We then develop the necessary geometric preliminaries with example calculations. We then use our newly developed tools to prove the local Gauss- Bonnet theorem. After a foray into triangulation and the Euler characteristic, we will use the local Gauss-Bonnet theorem to prove the global Gauss-Bonnet theorem for compact surfaces. Contents 1. Introduction 1 2. Gauss-Bonnet for particular triangles 2 2.1. Zero Gaussian Curvature 2 2.2. Constant Positive Gaussian Curvature 3 3. Geometric Preliminaries 3 3.1. Curves 3 3.2. Surfaces 4 3.3. The Tangent Plane 5 3.4. The First Fundamental Form 5 3.5. The Gauss Map and the Second Fundamental Form 7 3.6. Curvature(s) 8 3.7. The Covariant Derivative and Geodesics 11 3.8. Three Lemmas 12 4. Local Gauss-Bonnet Theorem 14 5. Global Gauss-Bonnet Theorem 15 Acknowledgments 17 References 17 1. Introduction The Gauss-Bonnet theorem serves as a fundamental connection between topol- ogy and geometry. It relates an inherently topological quanitity of a surface, the Euler characteristic, with an intrinsic geometric property, the total Gaussian cur- vature. Gaussian curvature is, roughly speaking, how much a surface deviates from a tangent plane at a point. The flatter the surface, the closer a tangent plane approximates it, and the lower the Gaussian curvature. The Gauss-Bonnet theo- rem remarkably states that under homeomorphism, the total Gaussian curvature remains invariant. This seems quite odd upon first glance as twisting, squishing, or 1 2 DUSTIN BURDA stretching an object seems like it would radically alter many of its geometric prop- erties. Yet, since the Euler characteric remains invaraiant under homeomorphism, the Gauss-Bonnet theorem predicts that total curvature should remain unchanged. Our goal is to build the necessary machinery to understand the theorem and then prove it. It formally states: Theorem 1.1. Let R ⊂ S be a regular region of an oriented surface and let C1 ··· Cn be the closed, simple, piecewise regular curves which form the boundary @R of R. Suppose that each Ci is positively oriented and let θ1 ··· θp be the set of all external angles of the curve C1 ··· Cn . Then n p X Z ZZ X kg(s)ds + Kdσ + θl = 2πχ(R); i=1 ci R l=1 where s denotes the arc length of Ci, and the integral over Ci means the sum of integrals in every regular arc of Ci. χ refers to the Euler characteristic and K refer to the total Gaussian curvature. The importance of Gauss-Bonnet is the connection it makes between these seem- ingly unrelated values. We will motivate both of these notions later in the paper and supply the requisite definitions. 2. Gauss-Bonnet for particular triangles The first version of the Gauss-Bonnet theorem that we will discuss concerns itself with geodesic triangles on a surface. It states that the difference between the angles of the triangle and π equals to the total Gaussian curvature of the triangle or 3 X ZZ (2.1) φi − π = KdA: i=1 T Geodesics, roughly speaking, are the shortest path along a surface. On a plane they are lines, on a sphere they are great circles, etc. We can think of curvature as how fast a tangent plane to a given point deviates from a surface. This formula enables us to compute the total Gaussian curvature along the triangle in a relatively simple manner. As opposed to complex computation with integrals, we calculate the angles in a given geodesic triangle. 2.1. Zero Gaussian Curvature. In a plane, the value K, or the Gaussian curva- ture, is equal to 0. This implies that the value on the right side of (2.1) evaluates to 0. Additionally, geodesics along the plane are merely straight lines. This im- plies that a geodesic triangle in a surface of flat curvature is a planar triangle. Rearranging the terms yields 3 X φi = π: i=1 This reduces to the theorem that the sum of the angles in a planar triangle add up to π radians. ELEMENTARY DIFFERENTIAL GEOMETRY AND THE GAUSS-BONNET THEOREM 3 2.2. Constant Positive Gaussian Curvature. Our next example is the unit sphere, which has constant positive curvature. The Gaussian curvature of a unit sphere is 1, meaning the right side of (2.1) is ZZ (2.2) 1dA; T but this is the area of the region we are integrating, which in our case is the geodesic triangle. Hence it is sufficient to prove 3 X (2.3) φi − π = 4; i=1 where 4 refers to the area of the geodesic triangle. Proof. Suppose there exists a geodesic triangle with angles α, β, γ. For each pair of edges of the triangle, their extensions along great circles form a lune, the 3 di- mensional analogue of a sector of a circle. The surface area of a lune is proportional to the angle it makes divided by 2π. In this case, the surface area of a lune with α angle α is 2π S, where S is the surface area of the sphere. Now, for each angle in the geodesic triangle, there exists two corresponding lunes. Hence, there are 6 lunes total. Their union is the surface area of the sphere. However, the geodesic triangle and its antipodal image are overcounted an additional 2 times by each lune, thus the geodesic triangle is overcounted 4 times. We then obtain the following equation: α β γ 2( + + )S = S + 4∆: 2π 2π 2π However, S = 4π, the surface area of a unit sphere. Therefore, substitution and simplification yields: α + β + γ − π = ∆: 3. Geometric Preliminaries 3.1. Curves. Before we prove the local and global Gauss-Bonnet theorems, we need to construct some objects and the operations we wish to perform on them. Definition 3.1. A parametrized differentiable curve is a differentiable curve α:I ! R3 of an open interval I = (a,b) of the real line R into R3. Remark 3.2. For each t 2 I, we denote α at t by α(t) = (x(t), y(t), z(t)) for some parametrization of the curve. Definition 3.3. The tangent vector α'(t) is the vector (x0(t), y0(t), z0(t)) Example 3.4. The map α(t) = (cos 2t, sin 4t, t2), where t 2 R is a parametrized differentiable curve. Each component is differentiable and α is a map from R into R3. The tangent vector at a point t is α0(t) = (−2 sin(2t); 4 cos(4t); 2t) Definition 3.5. A parametrized differentiable curve α:I ! R3 is said to be regular if α'(t) 6= 0 for all t 2 I. A variety of analytic tools can be used to explore the geometry of curves: 4 DUSTIN BURDA (1) The first derivative is the tangent vector. (2) The magnitude of the second derivative is defined to be the curvature. It measures the rate of the change of the tangent line. (3) The plane determined by the unit vectors in the direction of the first and second derivatives is the osculating plane at a point. (4) The measure of how quickly neighboring osculating planes differ from a given one is torsion. Curvature and torsion roughly correspond to the bending and twisting of a curve. Therefore we can think of any curve as bending, twisting, and stretching an interval. These serve as examples of how we can use analysis to explore geometry. Corresponding notions of bending and twisting as well as geometric properites such as area and length will be constructed for regular surfaces using the same idea. 3.2. Surfaces. We will now examine the notion of a surface. Surfaces are defined to be subsets rather than mappings. Definition 3.6. A subset S ⊂ R3 is a regular surface if, for each p 2 S, there exists a neighborhood V in R3 and a map x:U ! V \ S of an open set U ⊂ R2 onto V \ S such that: (1) x is smooth. (2) x is a homeomorphism . 2 3 (3) For each q 2 U, the differential dxq: R ! R is injective. Definition 3.7. The mapping x is called a parametrization of p. Definition 3.8. The neighborhood V \ S of p in S is called a coordinate neighborhood. In order to do differential geometry, we need all derivatives of x to be well- defined. This is guaranteed by Condition 1. Condition 2 enables our surface to be locally Euclidean, meaning it will inherit many of the same tools present in Eu- clidean space when viewed in a sufficently small neighborhood. Condition 3 takes on a more familiar form if we compute its matrix in the canonical basis of R2 and R3. 2 3 Let dxq have coordinates (u,v) in R and (x; y; z) in R . The vector e1 is tangent to the curve u ! (u, v0). It's image under x is (x(u; v0); y(u; v0); z(u; v0)). The tangent vector to the curve at x(q) is @x = @x @y @z : @u @u @u @u Thus, by the defintion of differential: @x dx (e ) = @x @y @z = ; q 1 @u @u @u @u hence 0 @x @x 1 @u @v B C dx = B @y @y C : q B @u @v C @ A @z @z @u @v ELEMENTARY DIFFERENTIAL GEOMETRY AND THE GAUSS-BONNET THEOREM 5 Condition 3 states that the two columns of the matrix of dxq are linearly inde- pendent.
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]
  • 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]
  • 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]
  • 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]
  • 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]
  • Connections and Curvature
    C Connections and Curvature Introduction In this appendix we present results in differential geometry that serve as a useful background for material in the main body of the book. Material in 1 on connections is somewhat parallel to the study of the natural connec- tion§ on a Riemannian manifold made in 11 of Chapter 1, but here we also study the curvature of a connection. Material§ in 2 on second covariant derivatives is connected with material in Chapter 2 on§ the Laplace operator. Ideas developed in 3 and 4, on the curvature of Riemannian manifolds and submanifolds, make§§ contact with such material as the existence of com- plex structures on two-dimensional Riemannian manifolds, established in Chapter 5, and the uniformization theorem for compact Riemann surfaces and other problems involving nonlinear, elliptic PDE, arising from studies of curvature, treated in Chapter 14. Section 5 on the Gauss-Bonnet theo- rem is useful both for estimates related to the proof of the uniformization theorem and for applications to the Riemann-Roch theorem in Chapter 10. Furthermore, it serves as a transition to more advanced material presented in 6–8. In§§ 6 we discuss how constructions involving vector bundles can be de- rived§ from constructions on a principal bundle. In the case of ordinary vector fields, tensor fields, and differential forms, one can largely avoid this, but it is a very convenient tool for understanding spinors. The principal bundle picture is used to construct characteristic classes in 7. The mate- rial in these two sections is needed in Chapter 10, on the index§ theory for elliptic operators of Dirac type.
    [Show full text]
  • Curvature Notions on Graphs Leeds Summer School, 18-19 July 2019
    CURVATURE NOTIONS ON GRAPHS LEEDS SUMMER SCHOOL, 18-19 JULY 2019 NORBERT PEYERIMHOFF Abstract. In these lecture notes we introduce some curvature notions on graphs. After a brief discussion of Gaussian curvature of surfaces we introduce three curvature notions which can be de- fined in the discrete setting of graphs: combinatorial curvature of surface tessellations, Ollivier Ricci curvature based on notions of Optimal Transport Theory and, finally, Gromov hyperbolicity cap- turing global aspects of negative curvature. Problems inserted in the text help to become more familiar with these curvature notions and to learn more about them. Contents 1. Introduction 1 2. Combinatorial curvature 7 3. Ollivier Ricci curvature 12 4. Gromov hyperbolicity 28 5. Summary 37 References 39 1. Introduction Curvature is a fundamental concept in Differential Geometry. In the case of surfaces in 3-dimensional Euclidean space, the Gaussian curvature at a given point of a surface is the product κ1κ2 of the prin- cipal curvatures (with signs corresponding to a given normal vector N). These principal curvatures are determined by two (orthogonal) direc- tions X1,X2, in which the surface bends most. In other words, we take the affine plane spanned by a tangent vector X and N and intersect this plane with the surface. The result is a curve which can be approx- imated, up to second order, at the point of the surface by a circle. The reciprocal value of the radius of this circle is the normal curvature in direction X. Rotating this tangent direction X leads to two orthogonal directions in which this normal curvature assumes extremal values.
    [Show full text]
  • 8. the Fary-Milnor Theorem
    Math 501 - Differential Geometry Herman Gluck Tuesday April 17, 2012 8. THE FARY-MILNOR THEOREM The curvature of a smooth curve in 3-space is 0 by definition, and its integral w.r.t. arc length, (s) ds , is called the total curvature of the curve. According to Fenchel's Theorem, the total curvature of any simple closed curve in 3-space is 2 , with equality if and only if it is a plane convex curve. 1 According to the Fary-Milnor Theorem, if the simple closed curve is knotted, then its total curvature is > 4 . In 1949, when Fary and Milnor proved this celebrated theorem independently, Fary was 27 years old and Milnor, an undergraduate at Princeton, was 18. In these notes, we'll prove Fenchel's Theorem first, and then the Fary-Milnor Theorem. 2 Istvan Fary (1922-1984) in 1968 3 John Milnor (1931 - ) 4 Fenchel's Theorem. We consider a smooth closed curve : [0, L] R3 , parametrized by arc-length. In order to make use of the associated Frenet frame, we assume that the curvature is never zero. FENCHEL's THEOREM (1929?). The total curvature of a smooth simple closed curve in 3-space is 2 , with equality if and only if it is a plane convex curve. 5 Proof. In this first step, we will show that the total curvature is 2 . We start with the smooth simple closed curve , construct a tubular neighborhood of radius r about it, and focus on the toroidal surface S of this tube. 6 Taking advantage of our assumption that the curvature of our curve never vanishes, we have a well-defined Frenet frame T(s) , N(s) , B(s) at each point (s) .
    [Show full text]
  • Research Statement
    RESEARCH STATEMENT CHAO LI 1. Introduction My research interests lie within differential geometry, partial differential equations and geomet- ric measure theory, with a focus on variational methods in geometric problems. A major theme of my research is to understand geometric and topological structures of manifolds via investigating stationary points of related geometric functionals. An important geometric functional is the area (or volume) functional for hypersurfaces Σn−1 in a Riemannian manifold (M n; g). Suppose Σ carries a unit normal vector field N (that is, Σ is 1 two-sided). The first variation of area along a normal deformation fN, f 2 Cc (M), is Z δVΣ(f) = fH; Σ where H is the mean curvature of Σ ⊂ (M; g). Critical points of V are called minimal hypersurfaces, and they are characterized by H ≡ 0. The second variation of volume is given by Z Z 2 Σ 2 2 2 2 δ VΣ(f) = jr fj − jAΣj f − RicM (N; N)f = fJ(f); Σ Σ Σ where r is the gradient operator on Σ, AΣ is the second fundamental form, RicM is the Ricci Σ 2 curvature of M, and J = −∆ − jAΣj − RicM (N; N) is the Jacobi operator. Assume Σ is compact. By standard elliptic theory, the number of negative Dirichlet eigenvalues of J is finite, and is called the Morse index of Σ ⊂ (M; g). A minimal hypersurface is called stable, if its Morse index is zero. Many important problems in differential geometry require a deep understanding of M(M; g; I), the space of two-sided minimal hypersurfaces in (M; g) with Morse index bounded by I.
    [Show full text]
  • DIFFERENTIAL GEOMETRY: a First Course in Curves and Surfaces
    DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces Preliminary Version Fall, 2015 Theodore Shifrin University of Georgia Dedicated to the memory of Shiing-Shen Chern, my adviser and friend c 2015 Theodore Shifrin No portion of this work may be reproduced in any form without written permission of the author, other than duplication at nominal cost for those readers or students desiring a hard copy. CONTENTS 1. CURVES.................... 1 1. Examples, Arclength Parametrization 1 2. Local Theory: Frenet Frame 10 3. SomeGlobalResults 23 2. SURFACES: LOCAL THEORY . 35 1. Parametrized Surfaces and the First Fundamental Form 35 2. The Gauss Map and the Second Fundamental Form 44 3. The Codazzi and Gauss Equations and the Fundamental Theorem of Surface Theory 57 4. Covariant Differentiation, Parallel Translation, and Geodesics 66 3. SURFACES: FURTHER TOPICS . 79 1. Holonomy and the Gauss-Bonnet Theorem 79 2. An Introduction to Hyperbolic Geometry 91 3. Surface Theory with Differential Forms 101 4. Calculus of Variations and Surfaces of Constant Mean Curvature 107 Appendix. REVIEW OF LINEAR ALGEBRA AND CALCULUS . 114 1. Linear Algebra Review 114 2. Calculus Review 116 3. Differential Equations 118 SOLUTIONS TO SELECTED EXERCISES . 121 INDEX ................... 124 Problems to which answers or hints are given at the back of the book are marked with an asterisk (*). Fundamental exercises that are particularly important (and to which reference is made later) are marked with a sharp (]). August, 2015 CHAPTER 1 Curves 1. Examples, Arclength Parametrization We say a vector function f .a; b/ R3 is Ck (k 0; 1; 2; : : :) if f and its first k derivatives, f , f ,..., W ! D 0 00 f.k/, exist and are all continuous.
    [Show full text]