DOCSLIB.ORG

Differential Geometry of Surfaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Differential Geometry of Surfaces Differential Geometry of Surfaces Jordan Smith and Carlo Sequin´ ∗ CS Division, UC Berkeley 1 Introduction fundamental form IS. This measures the length of a tangent vector on a given parametric basis by examining the quantity T T . These are notes on differential geometry of surfaces based on read- · ing [Greiner et al. n. d.]. 2 2 T T = (Su Su) @u + 2 (Su Sv) @u@v + (Sv Sv) @v (2) · · · · 2 Differential Geometry of Surfaces = E@u2 + 2F @u@v + G@v2 (3) Su @u Differential geometry of a 2D manifold or surface embedded in 3D = @u @v Su Sv (4) Sv · @v is the study of the intrinsic properties of the surface as well as the ef- fects of a given parameterization on the surface. For the discussion Su Su Su Sv @u = @u @v · · (5) below, we will consider the local behavior of a surface S at a single Su Sv Sv Sv @v point p with a given local parameterization (u; v), see Figure 1. We · · E F @u would like to calculate the principle curvature values (κ1; κ2), the @u @v (6) = F G @v maximum and minimum curvature values at p, as well as the local t orthogonal arc length parameterization (s; t) which is aligned with = U ISU (7) the directions of maximum and minimum curvature. Equation 7 defines IS. N Su S IS = gij = Su Sv (8) v Sv · Su Su Su Sv T = · · (9) Su Sv Sv Sv S · · u E F (10) = F G If an orthonormal arc length parameterization (s; t) is chosen then there will be no local metric distortion and IS will reduce to the identity matrix. Figure 1: Tangent plane of S 4 The Second Fundamental Form 3 The First Fundamental Form The unit normal N of a surface S at p is the vector perpendicular With the given parameterization, we can compute a pair of tangent to S, i.e. the tangent plane of S, at p. N can be calculated given a @S @S general nondegenerate parametrization (u; v). vectors Su Sv = @u @v assuming the u and v direc- tions are distinct. These two vectors locally parameterize the tan- gent plane to S at p.A general tangent vector T can be constructed Su Sv as a linear combination of Su Sv scaled by infinitesimal co- N = × (11) Su Sv @u @v t k × k efficients U = . The curvature of a surface S at a point p is defined by the rate @u T @uS @vS Su Sv (1) that N rotates in response to a unit tangential displacement as in = u + v = @v Figure 2. A normal section curve at p is constructed by intersecting Consider rotating the direction of the tangent T around p by set- S with a plane normal to it, i.e a plane that contains N and a tan- ting @u @v = cos θ sin θ . In general, the magnitude gent direction T . The curvature of this curve is the curvature of S in and direction of T will vary based on the skew and scale of the the direction T . The curvature κ of a curve is the reciprocal of the radius ρ of the best fitting osculating circle, i.e. κ = 1 . The direc- basisvectors Su Sv , see Figure 3(a). The distortion of the ρ local parameterization is described by the metric tensor or the first tional derivative NT of N in the direction T is a linear combination @N @N of the partial derivative vectors Nu Nv = @u @v ∗e-mail: (jordansjsequin)@cs.berkeley.edu and is parallel to the tangent plane. −1 Su (Suu Sv) + (Su Svu) @ Su Sv IIS = hij = Nu Nv (19) Nu = × × + (Su Sv) k × k Sv · − − Su Sv × @u k × k (12) Su Nu Su Nv = − · − · (20) −1 Sv Nu Sv Nv (Suv Sv) + (Su Svv) @ Su Sv − · − · Nv = × × + (Su Sv) k × k Su·(Suu×Sv) Su·(Suv×Sv) Su Sv × @u u v u v = − kS ×S k − kS ×S k (21) k × k (13) Sv·(Su×Svu) Sv·(Su×Svv ) " − kSu×Sv k − kSu×Sv k # @u Suu·(Su×Sv) Suv ·(Su×Sv) N = @uN + @vN = Nu Nv (14) T u v @v = kSu×Svk kSu×Svk (22) Svu·(Su×Sv ) Svv ·(Su×Sv) " kSu×Svk kSu×Svk # Suu N Suv N = · · (23) Suv N Svv N · · L M (24) N T = M N N Equations [23,24] can be substituted for IIS to yield alternate formulas for T NT . − · Sv 2 2 T NT = (Suu N) @u + 2 (Suv N) @u@v + (Svv N) @v T − · · · · (25) Su = L@u2 + 2M@u@v + N@v2 (26) Suu N Suv N @u = @u @v · · Svu N Svv N @v · · (27) L M @u = @u @v (28) M N @v Figure 2: Normal section of S IIS contains information about the curvature of S, but it will As with the tangent vector T , NT is subject to scaling by the be distorted by the metric if the given parameterization is not an metric of the parameter space. In an arc length parameter direc- arc length parameterization. It is still necessary to counter act this tion s, Ss Ns is the curvature of the normal section. Given a distortion in order to compute the curvature of the S. − · general parameterization (u; v), we can compute T NT . This − · quantity defines the second fundamental form IIS which describes curvature information distorted by the local metric. 5 Coordinate Transformations Our purpose in studying differential geometry and the fundamental forms has been to compute the principle curvature values (κ1; κ2) 2 T NT = (Su Nu) @u (Su Nv + Sv Nu) @u@v and directions (Sk1; Sk2) of a parametric surface S at a point p. − · − · − · · 2 Thus far we have derived IIS which contains curvature informa- (Sv Nv ) @v (15) − · tion but can be distorted by the metric of the parameterization. IS Su @u describes this metric information. We must combine IIS and IS to = @u @v − Nu Nv find the arc length curvature values. Sv · @v − The given parameterization u; v defines a set of basis tan- (16) ( ) gent vectors Su Sv . We can construct an orthonormal basis Su Nu Su Nv @u = @u @v − · − · Ss St due to an arc length parameterization (s; t) where Ss Sv Nu Sv Nv @v − · − · and Su are aligned, see Figure 3(a). A point T can be expressed in (17) either coordinate system, see Figure 3(b). t = U IISU (18) S S T = @u @v u = @s @t s (29) Sv St Equation 18 defines IIS. IIS can be simplified to a set of quanties which are easier to compute by substituting the expres- sions in Equations [12,13] for Nu and Nv respectively. Two of the The coordinates @s @t measure the length of T as op- terms of in of the entries of the matrix in Equation 21 vanish due to posed to the coordinates @u @v . the dot product of orthogonal vectors. Further simplification using properties of the box product of vectors yields Equation 22. Equa- T T =@s2 + @t2 = @u2 + @v2 (30) tion 23 is derived by subsituting N from Equation 11 and shows · 6 We would like to work in the Ss St , so we must find the that IIS can be computed simply by the dot product of the second transformations to and from S S . The quantities a, b, and partial derivatives of S with N. Equation 23 also demonstrates that u v II is a symmetric matrix. θ in Figure 3(a) are defined by S S . S u v St Su @u T T = @u @v Su Sv (42) · Sv · @v @u Sv @u @v I (43) = S @v s u T S b t −1 −1 t @s v = @s @t A I A (44) S @t S v t θ b t @s = @s @t A−1 AAt A−1 (45) Ss θ @t a a S S S u u s @s @s @t (46) = @t (a) Constructing an or- (b) A point T in (u; v) and thonormal basis on top of (s; t) coordinates = @ s2 + @t2 (47) the given parametric basis It is now possible to transform our given parameters to coordi- nates on an orthonormal basis where we can measure lengths. We Figure 3: Effects of the metric can also transform coordinates on the orthonormal basis back to coordinates on our given parameterization. 6 Curvature The curvature at a point p on the surface S in any tangential direc- Su a 0 Ss tion T is T N . = (31) T Sv b cos θ b sin θ St − · S = A s (32) Su @u S T NT = @u @v Nu Nv t − · Sv · − − @v S 1 b sin θ 0 S (48) s = u (33) St ab sin θ b cos θ a Sv @u − = @u @v IIS (49) S @v = A−1 u (34) Sv This function is parameterized by the given parameterization. We would like to study the function that sweeps a unit length tan- gent around p, so it is more convenient to use arc length coordinates. The first fundamental form IS can be represented by these coor- dinate transformations. We must transform coordinates so that setting the arc length coordi- nates @s @t = cos φ sin φ sweeps out the curvature values. Su IS = Su Sv (35) −1 −1 t @s Sv · T N = @s @t A II A (50) T S @t − · Ss t = A Ss St A (36) @s St · = @s @t II (51) S^ @t 1 0 t = A A (37) 0 1 S = AAt (38) t E1 a2 ab cos θ = 2 (39) S ab cos θ b E v 2 b We can also derive transformations between coordinates of the φ different sets of basis vectors by substituting Equations [32,34] re- θ S spectively into Equation 29.
Load more

Recommended publications
  • An Introduction to Topology the Classification Theorem for Surfaces by E
    An Introduction to Topology An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman Introduction. The classification theorem is a beautiful example of geometric topology. Although it was discovered in the last century*, yet it manages to convey the spirit of present day research. The proof that we give here is elementary, and its is hoped more intuitive than that found in most textbooks, but in none the less rigorous. It is designed for readers who have never done any topology before. It is the sort of mathematics that could be taught in schools both to foster geometric intuition, and to counteract the present day alarming tendency to drop geometry. It is profound, and yet preserves a sense of fun. In Appendix 1 we explain how a deeper result can be proved if one has available the more sophisticated tools of analytic topology and algebraic topology. Examples. Before starting the theorem let us look at a few examples of surfaces. In any branch of mathematics it is always a good thing to start with examples, because they are the source of our intuition. All the following pictures are of surfaces in 3-dimensions. In example 1 by the word “sphere” we mean just the surface of the sphere, and not the inside. In fact in all the examples we mean just the surface and not the solid inside. 1. Sphere. 2. Torus (or inner tube). 3. Knotted torus. 4. Sphere with knotted torus bored through it. * Zeeman wrote this article in the mid-twentieth century. 1 An Introduction to Topology 5.
    [Show full text]
  • Surface Physics I and II
    Surface physics I and II Lectures: Mon 12-14 ,Wed 12-14 D116 Excercises: TBA Lecturer: Antti Kuronen, [email protected] Exercise assistant: Ane Lasa, [email protected] Course homepage: http://www.physics.helsinki.fi/courses/s/pintafysiikka/ Objectives ● To study properties of surfaces of solid materials. ● The relationship between the composition and morphology of the surface and its mechanical, chemical and electronic properties will be dealt with. ● Technologically important field of surface and thin film growth will also be covered. Surface physics I 2012: 1. Introduction 1 Surface physics I and II ● Course in two parts ● Surface physics I (SPI) (530202) ● Period III, 5 ECTS points ● Basics of surface physics ● Surface physics II (SPII) (530169) ● Period IV, 5 ECTS points ● 'Special' topics in surface science ● Surface and thin film growth ● Nanosystems ● Computational methods in surface science ● You can take only SPI or both SPI and SPII Surface physics I 2012: 1. Introduction 2 How to pass ● Both courses: ● Final exam 50% ● Exercises 50% ● Exercises ● Return by ● email to [email protected] or ● on paper to course box on the 2nd floor of Physicum ● Return by (TBA) Surface physics I 2012: 1. Introduction 3 Table of contents ● Surface physics I ● Introduction: What is a surface? Why is it important? Basic concepts. ● Surface structure: Thermodynamics of surfaces. Atomic and electronic structure. ● Experimental methods for surface characterization: Composition, morphology, electronic properties. ● Surface physics II ● Theoretical and computational methods in surface science: Analytical models, Monte Carlo and molecular dynamics sumilations. ● Surface growth: Adsorption, desorption, surface diffusion. ● Thin film growth: Homoepitaxy, heteroepitaxy, nanostructures.
    [Show full text]
  • Surface Water
    Chapter 5 SURFACE WATER Surface water originates mostly from rainfall and is a mixture of surface run-off and ground water. It includes larges rivers, ponds and lakes, and the small upland streams which may originate from springs and collect the run-off from the watersheds. The quantity of run-off depends upon a large number of factors, the most important of which are the amount and intensity of rainfall, the climate and vegetation and, also, the geological, geographi- cal, and topographical features of the area under consideration. It varies widely, from about 20 % in arid and sandy areas where the rainfall is scarce to more than 50% in rocky regions in which the annual rainfall is heavy. Of the remaining portion of the rainfall. some of the water percolates into the ground (see "Ground water", page 57), and the rest is lost by evaporation, transpiration and absorption. The quality of surface water is governed by its content of living organisms and by the amounts of mineral and organic matter which it may have picked up in the course of its formation. As rain falls through the atmo- sphere, it collects dust and absorbs oxygen and carbon dioxide from the air. While flowing over the ground, surface water collects silt and particles of organic matter, some of which will ultimately go into solution. It also picks up more carbon dioxide from the vegetation and micro-organisms and bacteria from the topsoil and from decaying matter. On inhabited watersheds, pollution may include faecal material and pathogenic organisms, as well as other human and industrial wastes which have not been properly disposed of.
    [Show full text]
  • 3D Modeling: Surfaces
    CS 430/536 Computer Graphics I Overview • 3D model representations 3D Modeling: • Mesh formats Surfaces • Bicubic surfaces • Bezier surfaces Week 8, Lecture 16 • Normals to surfaces David Breen, William Regli and Maxim Peysakhov • Direct surface rendering Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University 1 2 http://gicl.cs.drexel.edu 1994 Foley/VanDam/Finer/Huges/Phillips ICG 3D Modeling Representing 3D Objects • 3D Representations • Exact • Approximate – Wireframe models – Surface Models – Wireframe – Facet / Mesh – Solid Models – Parametric • Just surfaces – Meshes and Polygon soups – Voxel/Volume models Surface – Voxel – Decomposition-based – Solid Model • Volume info • Octrees, voxels • CSG • Modeling in 3D – Constructive Solid Geometry (CSG), • BRep Breps and feature-based • Implicit Solid Modeling 3 4 Negatives when Representing 3D Objects Representing 3D Objects • Exact • Approximate • Exact • Approximate – Complex data structures – Lossy – Precise model of – A discretization of – Expensive algorithms – Data structure sizes can object topology the 3D object – Wide variety of formats, get HUGE, if you want each with subtle nuances good fidelity – Mathematically – Use simple – Hard to acquire data – Easy to break (i.e. cracks represent all primitives to – Translation required for can appear) rendering – Not good for certain geometry model topology applications • Lots of interpolation and and geometry guess work 5 6 1 Positives when Exact Representations Representing 3D Objects • Exact
    [Show full text]
  • Surface Topology
    2 Surface topology 2.1 Classification of surfaces In this second introductory chapter, we change direction completely. We dis- cuss the topological classification of surfaces, and outline one approach to a proof. Our treatment here is almost entirely informal; we do not even define precisely what we mean by a ‘surface’. (Definitions will be found in the following chapter.) However, with the aid of some more sophisticated technical language, it not too hard to turn our informal account into a precise proof. The reasons for including this material here are, first, that it gives a counterweight to the previous chapter: the two together illustrate two themes—complex analysis and topology—which run through the study of Riemann surfaces. And, second, that we are able to introduce some more advanced ideas that will be taken up later in the book. The statement of the classification of closed surfaces is probably well known to many readers. We write down two families of surfaces g, h for integers g ≥ 0, h ≥ 1. 2 2 The surface 0 is the 2-sphere S . The surface 1 is the 2-torus T .For g ≥ 2, we define the surface g by taking the ‘connected sum’ of g copies of the torus. In general, if X and Y are (connected) surfaces, the connected sum XY is a surface constructed as follows (Figure 2.1). We choose small discs DX in X and DY in Y and cut them out to get a pair of ‘surfaces-with- boundaries’, coresponding to the circle boundaries of DX and DY.
    [Show full text]
  • Surface Integrals
    VECTOR CALCULUS 16.7 Surface Integrals In this section, we will learn about: Integration of different types of surfaces. PARAMETRIC SURFACES Suppose a surface S has a vector equation r(u, v) = x(u, v) i + y(u, v) j + z(u, v) k (u, v) D PARAMETRIC SURFACES •We first assume that the parameter domain D is a rectangle and we divide it into subrectangles Rij with dimensions ∆u and ∆v. •Then, the surface S is divided into corresponding patches Sij. •We evaluate f at a point Pij* in each patch, multiply by the area ∆Sij of the patch, and form the Riemann sum mn * f() Pij S ij ij11 SURFACE INTEGRAL Equation 1 Then, we take the limit as the number of patches increases and define the surface integral of f over the surface S as: mn * f( x , y , z ) dS lim f ( Pij ) S ij mn, S ij11 . Analogues to: The definition of a line integral (Definition 2 in Section 16.2);The definition of a double integral (Definition 5 in Section 15.1) . To evaluate the surface integral in Equation 1, we approximate the patch area ∆Sij by the area of an approximating parallelogram in the tangent plane. SURFACE INTEGRALS In our discussion of surface area in Section 16.6, we made the approximation ∆Sij ≈ |ru x rv| ∆u ∆v where: x y z x y z ruv i j k r i j k u u u v v v are the tangent vectors at a corner of Sij. SURFACE INTEGRALS Formula 2 If the components are continuous and ru and rv are nonzero and nonparallel in the interior of D, it can be shown from Definition 1—even when D is not a rectangle—that: fxyzdS(,,) f ((,))|r uv r r | dA uv SD SURFACE INTEGRALS This should be compared with the formula for a line integral: b fxyzds(,,) f (())|'()|rr t tdt Ca Observe also that: 1dS |rr | dA A ( S ) uv SD SURFACE INTEGRALS Example 1 Compute the surface integral x2 dS , where S is the unit sphere S x2 + y2 + z2 = 1.
    [Show full text]
  • 16.6 Parametric Surfaces and Their Areas
    CHAPTER 16. VECTOR CALCULUS 239 16.6 Parametric Surfaces and Their Areas Comments. On the next page we summarize some ways of looking at graphs in Calc I, Calc II, and Calc III, and pose a question about what comes next. 2 3 2 Definition. Let r : R ! R and let D ⊆ R . A parametric surface S is the set of all points hx; y; zi such that hx; y; zi = r(u; v) for some hu; vi 2 D. In other words, it's the set of points you get using x = some function of u; v y = some function of u; v z = some function of u; v Example 1. (a) Describe the surface z = −x2 − y2 + 2 over the region D : −1 ≤ x ≤ 1; −1 ≤ y ≤ 1 as a parametric surface. Graph it in MATLAB. (b) Describe the surface z = −x2 − y2 + 2 over the region D : x2 + y2 ≤ 1 as a parametric surface. Graph it in MATLAB. Solution: (a) In this case we \cheat": we already have z as a function of x and y, so anything we do to get rid of x and y will work x = u y = v z = −u2 − v2 + 2 −1 ≤ u ≤ 1; −1 ≤ v ≤ 1 [u,v]=meshgrid(linspace(-1,1,35)); x=u; y=v; z=2-u.^2-v.^2; surf(x,y,z) CHAPTER 16. VECTOR CALCULUS 240 How graphs look in different contexts: Pre-Calc, and Calc I graphs: Calc II parametric graphs: y = f(x) x = f(t), y = g(t) • 1 number gets plugged in (usually x) • 1 number gets plugged in (usually t) • 1 number comes out (usually y) • 2 numbers come out (x and y) • Graph is essentially one dimensional (if you zoom in enough it looks like a line, plus you only need • graph is essentially one dimensional (but we one number, x, to specify any location on the don't picture t at all) graph) • We need to picture the graph in two dimensional • We need to picture the graph in a two dimen- world.
    [Show full text]
  • Polynomial Curves and Surfaces
    Polynomial Curves and Surfaces Chandrajit Bajaj and Andrew Gillette September 8, 2010 Contents 1 What is an Algebraic Curve or Surface? 2 1.1 Algebraic Curves . .3 1.2 Algebraic Surfaces . .3 2 Singularities and Extreme Points 4 2.1 Singularities and Genus . .4 2.2 Parameterizing with a Pencil of Lines . .6 2.3 Parameterizing with a Pencil of Curves . .7 2.4 Algebraic Space Curves . .8 2.5 Faithful Parameterizations . .9 3 Triangulation and Display 10 4 Polynomial and Power Basis 10 5 Power Series and Puiseux Expansions 11 5.1 Weierstrass Factorization . 11 5.2 Hensel Lifting . 11 6 Derivatives, Tangents, Curvatures 12 6.1 Curvature Computations . 12 6.1.1 Curvature Formulas . 12 6.1.2 Derivation . 13 7 Converting Between Implicit and Parametric Forms 20 7.1 Parameterization of Curves . 21 7.1.1 Parameterizing with lines . 24 7.1.2 Parameterizing with Higher Degree Curves . 26 7.1.3 Parameterization of conic, cubic plane curves . 30 7.2 Parameterization of Algebraic Space Curves . 30 7.3 Automatic Parametrization of Degree 2 Curves and Surfaces . 33 7.3.1 Conics . 34 7.3.2 Rational Fields . 36 7.4 Automatic Parametrization of Degree 3 Curves and Surfaces . 37 7.4.1 Cubics . 38 7.4.2 Cubicoids . 40 7.5 Parameterizations of Real Cubic Surfaces . 42 7.5.1 Real and Rational Points on Cubic Surfaces . 44 7.5.2 Algebraic Reduction . 45 1 7.5.3 Parameterizations without Real Skew Lines . 49 7.5.4 Classification and Straight Lines from Parametric Equations . 52 7.5.5 Parameterization of general algebraic plane curves by A-splines .
    [Show full text]
  • Differential Geometry of Curves and Surfaces 3
    DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 3. Regular Surfaces 3.1. The definition of a regular surface. Examples. The notion of surface we are going to deal with in our course can be intuitively understood as the object obtained by a potter full of phantasy who takes several pieces of clay, flatten them on a table, then models of each of them arbitrarily strange looking pots, and finally glue them together and flatten the possible edges and corners. The resulting object is, formally speaking, a subset of the three dimensional space R3. Here is the rigorous definition of a surface. Definition 3.1.1. A subset S ⊂ R3 is a1 regular surface if for any point P in S one can find an open subspace U ⊂ R2 and a map ϕ : U → S of the form ϕ(u, v)=(x(u, v),y(u, v), z(u, v)), for (u, v) ∈ U with the following properties: 1. The function ϕ is differentiable, in the sense that the components x(u, v), y(u, v) and z(u, v) have partial derivatives of any order. 2 3 2. For any Q in U, the differential map d(ϕ)Q : R → R is (linear and) injective. 3. There exists an open subspace V ⊂ R3 which contains P such that ϕ(U) = V ∩ S and moreover, ϕ : U → V ∩ S is a homeomorphism2. The pair (U, ϕ) is called a local parametrization, or a chart, or a local coordinate system around P . Conditions 1 and 2 say that (U, ϕ) is a regular patch.
    [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]
  • 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]
  • Convexity, Rigidity, and Reduction of Codimension of Isometric
    CONVEXITY, RIGIDITY, AND REDUCTION OF CODIMENSION OF ISOMETRIC IMMERSIONS INTO SPACE FORMS RONALDO F. DE LIMA AND RUBENS L. DE ANDRADE To Elon Lima, in memoriam, and Manfredo do Carmo, on his 90th birthday Abstract. We consider isometric immersions of complete connected Riemann- ian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding with codimension one, its image bounds a convex set, and it is rigid. This result generalizes previous ones by M. do Carmo and E. Lima, as well as by M. do Carmo and F. Warner. It also settles affirma- tively a conjecture by do Carmo and Warner. We establish a similar result for complete isometric immersions satisfying a stronger condition on the second fundamental form. We extend to the context of isometric immersions in space forms a classical theorem for Euclidean hypersurfaces due to Hadamard. In this same context, we prove an existence theorem for hypersurfaces with pre- scribed boundary and vanishing Gauss-Kronecker curvature. Finally, we show that isometric immersions into space forms which are regular outside the set of totally geodesic points admit a reduction of codimension to one. 1. Introduction Convexity and rigidity are among the most essential concepts in the theory of submanifolds. In his work, R. Sacksteder established two fundamental results in- volving these concepts. Combined, they state that for M n a non-flat complete connected Riemannian manifold with nonnegative sectional curvatures, any iso- metric immersion f : M n Rn+1 is, in fact, an embedding and has f(M) as the boundary of a convex set.
    [Show full text]