DOCSLIB.ORG

DIFFERENTIAL GEOMETRY: a First Course in Curves and Surfaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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. We say f is smooth if f is Ck for every positive integer k.A parametrized curve is a C3 (or smooth) map ˛ I R3 for some interval I .a; b/ or Œa; b in R (possibly infinite). We W ! D say ˛ is regular if ˛ .t/ 0 for all t I . 0 ¤ 2 We can imagine a particle moving along the path ˛, with its position at time t given by ˛.t/. As we learned in vector calculus, d˛ ˛.t h/ ˛.t/ ˛0.t/ lim C D dt D h 0 h ! is the velocity of the particle at time t. The velocity vector ˛0.t/ is tangent to the curve at ˛.t/ and its length, ˛ .t/ , is the speed of the particle. k 0 k Example 1. We begin with some standard examples. (a) Familiar from linear algebra and vector calculus is a parametrized line: Given points P and Q in R3, we let v !PQ Q P and set ˛.t/ P tv, t R. Note that ˛.0/ P , ˛.1/ Q, D D D C 2 D D and for 0 t 1, ˛.t/ is on the line segment PQ. We ask the reader to check in Exercise 8 that of all paths from P to Q, the “straight line path” ˛ gives the shortest. This is typical of problems we shall consider in the future. (b) Essentially by the very definition of the trigonometric functions cos and sin, we obtain a very natural parametrization of a circle of radius a, as pictured in Figure 1.1(a): ˛.t/ a cos t; sin t a cos t; a sin t ; 0 t 2: D D (a cos t, a sin t) (a cos t, b sin t) b t a a (a) (b) F IGURE 1.1 1 2 CHAPTER 1. CURVES (c) Now, if a; b > 0 and we apply the linear map T R2 R2; T .x; y/ .ax; by/; W ! D we see that the unit circle x2 y2 1 maps to the ellipse x2=a2 y2=b2 1. Since T.cos t; sin t/ C D C D D .a cos t; b sin t/, the latter gives a natural parametrization of the ellipse, as shown in Figure 1.1(b). (d) Consider the two cubic curves in R2 illustrated in Figure 1.2. On the left is the cuspidal cubic y=tx y2=x3+x2 y2=x3 (a) (b) F IGURE 1.2 y2 x3, and on the right is the nodal cubic y2 x3 x2. These can be parametrized, respectively, D D C by the functions ˛.t/ .t2; t3/ and ˛.t/ .t2 1; t.t2 1//: D D (In the latter case, as the figure suggests, we see that the line y tx intersects the curve when D .tx/2 x2.x 1/, so x 0 or x t2 1.) D C D D z2=y3 z=x3 y=x2 F IGURE 1.3 .E÷1 XAMPLES,ARCLENGTH PARAMETRIZATION 3 (e) Now consider the twisted cubic in R3, illustrated in Figure 1.3, given by ˛.t/ .t; t2; t3/; t R: D 2 Its projections in the xy-, xz-, and yz-coordinate planes are, respectively, y x2, z x3, and D D z2 y3 (the cuspidal cubic). D (f) Our next example is a classic called the cycloid: It is the trajectory of a dot on a rolling wheel (circle). Consider the illustration in Figure 1.4. Assuming the wheel rolls without slipping, the P t a O F IGURE 1.4 distance it travels along the ground is equal to the length of the circular arc subtended by the angle through which it has turned. That is, if the radius of the circle is a and it has turned through angle t, then the point of contact with the x-axis, Q, is at units to the right. The vector from the origin to C C a P t a cos t P a sin t O Q F IGURE 1.5 the point P can be expressed as the sum of the three vectors !OQ, !QC , and !CP (see Figure 1.5): !OP !OQ !QC !CP D C C .at; 0/ .0; a/ . a sin t; a cos t/; D C C and hence the function ˛.t/ .at a sin t; a a cos t/ a.t sin t; 1 cos t/; t R D D 2 gives a parametrization of the cycloid. (g) A (circular) helix is the screw-like path of a bug as it walks uphill on a right circular cylinder at a constant slope or pitch. If the cylinder has radius a and the slope is b=a, we can imagine drawing a line of that slope on a piece of paper 2a units long, and then rolling the paper up into a cylinder. The line gives one revolution of the helix, as we can see in Figure 1.6. If we take the axis of the cylinder to be vertical, the projection of the helix in the horizontal plane is a circle of radius a, and so we obtain the parametrization ˛.t/ .a cos t; a sin t; bt/. D 4 CHAPTER 1. CURVES F IGURE 1.6 Brief review of hyperbolic trigonometric functions. Just as the circle x2 y2 1 is parametrized C D by .cos ; sin /, the portion of the hyperbola x2 y2 1 lying to the right of the y-axis, as shown D in Figure 1.7, is parametrized by .cosh t; sinh t/, where et e t et e t cosh t C and sinh t : D 2 D 2 sinh t 1 By analogy with circular trigonometry, we set tanh t and sech t . The following D cosh t D cosh t (cosh t, sinh t) F IGURE 1.7 formulas are easy to check: cosh2 t sinh2 t 1; tanh2 t sech2 t 1 D C D sinh .t/ cosh t; cosh .t/ sinh t; tanh .t/ sech2 t; sech .t/ tanh t sech t: 0 D 0 D 0 D 0 D .E÷1 XAMPLES,ARCLENGTH PARAMETRIZATION 5 (h) When a uniform and flexible chain hangs from two pegs, its weight is uniformly distributed along its length. The shape it takes is called a catenary.1 As we ask the reader to check in Exercise 9, the catenary is the graph of f .x/ C cosh.x=C /, for any constant C > 0. This curve will appear D F IGURE 1.8 numerous times in this course. O Example 2. One of the more interesting curves that arise “in nature” is the tractrix.2 The traditional story is this: A dog is at the end of a 1-unit leash and buries a bone at .0; 1/ as his owner begins to walk down the x-axis, starting at the origin. The dog tries to get back to the bone, so he always pulls the leash taut as he is dragged along the tractrix by his owner. His pulling the leash taut means that the leash will be tangent to the curve. When the master is at .t; 0/, let the dog’s position be .x.t/;y.t//, and let the leash F IGURE 1.9 make angle .t/ with the positive x-axis. Then we have x.t/ t cos .t/, y.t/ sin .t/, so D C D dy y .t/ cos .t/ .t/ tan .t/ 0 0 : D dx D x .t/ D 1 sin .t/ .t/ 0 0 Therefore, .t/ sin .t/. Separating variables and integrating, we have d= sin dt, and so 0 D D t ln.csc cot / c for some constant c. Since =2 when t 0, we see that c 0. Now, R R D C 1 Ccos 2 cos2.=2/ D D D since csc cot C cot.=2/, we can rewrite this as t ln tan.=2/. C D sin D 2 sin.=2/ cos.=2/ D D Thus, we can parametrize the tractrix by ˛./ cos ln tan.=2/; sin ; =2 < : D C 1From the Latin cat¯ena, chain.
Recommended publications
  • Engineering Curves – I

    Engineering Curves – I

    Engineering Curves – I 1. Classification 2. Conic sections - explanation 3. Common Definition 4. Ellipse – ( six methods of construction) 5. Parabola – ( Three methods of construction) 6. Hyperbola – ( Three methods of construction ) 7. Methods of drawing Tangents & Normals ( four cases) Engineering Curves – II 1. Classification 2. Definitions 3. Involutes - (five cases) 4. Cycloid 5. Trochoids – (Superior and Inferior) 6. Epic cycloid and Hypo - cycloid 7. Spiral (Two cases) 8. Helix – on cylinder & on cone 9. Methods of drawing Tangents and Normals (Three cases) ENGINEERING CURVES Part- I {Conic Sections} ELLIPSE PARABOLA HYPERBOLA 1.Concentric Circle Method 1.Rectangle Method 1.Rectangular Hyperbola (coordinates given) 2.Rectangle Method 2 Method of Tangents ( Triangle Method) 2 Rectangular Hyperbola 3.Oblong Method (P-V diagram - Equation given) 3.Basic Locus Method 4.Arcs of Circle Method (Directrix – focus) 3.Basic Locus Method (Directrix – focus) 5.Rhombus Metho 6.Basic Locus Method Methods of Drawing (Directrix – focus) Tangents & Normals To These Curves. CONIC SECTIONS ELLIPSE, PARABOLA AND HYPERBOLA ARE CALLED CONIC SECTIONS BECAUSE THESE CURVES APPEAR ON THE SURFACE OF A CONE WHEN IT IS CUT BY SOME TYPICAL CUTTING PLANES. OBSERVE ILLUSTRATIONS GIVEN BELOW.. Ellipse Section Plane Section Plane Hyperbola Through Generators Parallel to Axis. Section Plane Parallel to end generator. COMMON DEFINATION OF ELLIPSE, PARABOLA & HYPERBOLA: These are the loci of points moving in a plane such that the ratio of it’s distances from a fixed point And a fixed line always remains constant. The Ratio is called ECCENTRICITY. (E) A) For Ellipse E<1 B) For Parabola E=1 C) For Hyperbola E>1 Refer Problem nos.
  • Toponogov.V.A.Differential.Geometry

    Toponogov.V.A.Differential.Geometry

    Victor Andreevich Toponogov with the editorial assistance of Vladimir Y. Rovenski Differential Geometry of Curves and Surfaces A Concise Guide Birkhauser¨ Boston • Basel • Berlin Victor A. Toponogov (deceased) With the editorial assistance of: Department of Analysis and Geometry Vladimir Y. Rovenski Sobolev Institute of Mathematics Department of Mathematics Siberian Branch of the Russian Academy University of Haifa of Sciences Haifa, Israel Novosibirsk-90, 630090 Russia Cover design by Alex Gerasev. AMS Subject Classification: 53-01, 53Axx, 53A04, 53A05, 53A55, 53B20, 53B21, 53C20, 53C21 Library of Congress Control Number: 2005048111 ISBN-10 0-8176-4384-2 eISBN 0-8176-4402-4 ISBN-13 978-0-8176-4384-3 Printed on acid-free paper. c 2006 Birkhauser¨ Boston All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkhauser¨ Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA) and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and re- trieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (TXQ/EB) 987654321 www.birkhauser.com Contents Preface ....................................................... vii About the Author .............................................
  • Abstract Book

    Abstract Book

    14th International Geometry Symposium 25-28 May 2016 ABSTRACT BOOK Pamukkale University Denizli - TURKEY 1 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 14th International Geometry Symposium ABSTRACT BOOK 1 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 Proceedings of the 14th International Geometry Symposium Edited By: Dr. Şevket CİVELEK Dr. Cansel YORMAZ E-Published By: Pamukkale University Department of Mathematics Denizli, TURKEY All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means or whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder. Authors of papers in these proceedings are authorized to use their own material freely. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to: Assoc. Prof. Dr. Şevket CİVELEK Pamukkale University Department of Mathematics Denizli, TURKEY Email: [email protected] 2 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 Proceedings of the 14th International Geometry Symposium May 25-28, 2016 Denizli, Turkey. Jointly Organized by Pamukkale University Department of Mathematics Denizli, Turkey 3 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 PREFACE This volume comprises the abstracts of contributed papers presented at the 14th International Geometry Symposium, 14IGS 2016 held on May 25-28, 2016, in Denizli, Turkey. 14IGS 2016 is jointly organized by Department of Mathematics, Pamukkale University, Denizli, Turkey. The sysposium is aimed to provide a platform for Geometry and its applications.
  • Extended Rectifying Curves As New Kind of Modified Darboux Vectors

    Extended Rectifying Curves As New Kind of Modified Darboux Vectors

    TWMS J. Pure Appl. Math., V.9, N.1, 2018, pp.18-31 EXTENDED RECTIFYING CURVES AS NEW KIND OF MODIFIED DARBOUX VECTORS Y. YAYLI1, I. GOK¨ 1, H.H. HACISALIHOGLU˘ 1 Abstract. Rectifying curves are defined as curves whose position vectors always lie in recti- fying plane. The centrode of a unit speed curve in E3 with nonzero constant curvature and non-constant torsion (or nonzero constant torsion and non-constant curvature) is a rectifying curve. In this paper, we give some relations between non-helical extended rectifying curves and their Darboux vector fields using any orthonormal frame along the curves. Furthermore, we give some special types of ruled surface. These surfaces are formed by choosing the base curve as one of the integral curves of Frenet vector fields and the director curve δ as the extended modified Darboux vector fields. Keywords: rectifying curve, centrodes, Darboux vector, conical geodesic curvature. AMS Subject Classification: 53A04, 53A05, 53C40, 53C42 1. Introduction From elementary differential geometry it is well known that at each point of a curve α, its planes spanned by fT;Ng, fT;Bg and fN; Bg are known as the osculating plane, the rectifying plane and the normal plane, respectively. A curve called twisted curve has non-zero curvature functions in the Euclidean 3-space. Rectifying curves are introduced by B. Y. Chen in [3] as space curves whose position vector always lies in its rectifying plane, spanned by the tangent and the binormal vector fields T and B of the curve. Accordingly, the position vector with respect to some chosen origin, of a rectifying curve α in E3, satisfies the equation α(s) = λ(s)T (s) + µ(s)B(s) for some functions λ(s) and µ(s): He proved that a twisted curve is congruent to a rectifying τ curve if and only if the ratio { is a non-constant linear function of arclength s: Subsequently Ilarslan and Nesovic generalized the rectifying curves in Euclidean 3-space to Euclidean 4-space [10].
  • DISCRETE DIFFERENTIAL GEOMETRY: an APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 15: CURVATURE

    DISCRETE DIFFERENTIAL GEOMETRY: an APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 15: CURVATURE

    DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 LECTURE 15: CURVATURE DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane • CMU 15-458/858 Curvature—Overview • Intuitively, describes “how much a shape bends” – Extrinsic: how quickly does the tangent plane/normal change? – Intrinsic: how much do quantities differ from flat case? N T B Curvature—Overview • Driving force behind wide variety of physical phenomena – Objects want to reduce—or restore—their curvature – Even space and time are driven by curvature… Curvature—Overview • Gives a coordinate-invariant description of shape – fundamental theorems of plane curves, space curves, surfaces, … • Amazing fact: curvature gives you information about global topology! – “local-global theorems”: turning number, Gauss-Bonnet, … Curvature—Overview • Geometric algorithms: shape analysis, local descriptors, smoothing, … • Numerical simulation: elastic rods/shells, surface tension, … • Image processing algorithms: denoising, feature/contour detection, … Thürey et al 2010 Gaser et al Kass et al 1987 Grinspun et al 2003 Curvature of Curves Review: Curvature of a Plane Curve • Informally, curvature describes “how much a curve bends” • More formally, the curvature of an arc-length parameterized plane curve can be expressed as the rate of change in the tangent Equivalently: Here the angle brackets denote the usual dot product, i.e., . Review: Curvature and Torsion of a Space Curve •For a plane curve, curvature captured the notion of “bending” •For a space curve we also have torsion, which captures “twisting” Intuition: torsion is “out of plane bending” increasing torsion Review: Fundamental Theorem of Space Curves •The fundamental theorem of space curves tells that given the curvature κ and torsion τ of an arc-length parameterized space curve, we can recover the curve (up to rigid motion) •Formally: integrate the Frenet-Serret equations; intuitively: start drawing a curve, bend & twist at prescribed rate.
  • Extension of the Darboux Frame Into Euclidean 4-Space and Its Invariants

    Extension of the Darboux Frame Into Euclidean 4-Space and Its Invariants

    Turkish Journal of Mathematics Turk J Math (2017) 41: 1628 { 1639 http://journals.tubitak.gov.tr/math/ ⃝c TUB¨ ITAK_ Research Article doi:10.3906/mat-1604-56 Extension of the Darboux frame into Euclidean 4-space and its invariants Mustafa DULD¨ UL¨ 1;∗, Bahar UYAR DULD¨ UL¨ 2, Nuri KURUOGLU˘ 3, Ertu˘grul OZDAMAR¨ 4 1Department of Mathematics, Faculty of Science and Arts, Yıldız Technical University, Istanbul,_ Turkey 2Department of Mathematics Education, Faculty of Education, Yıldız Technical University, Istanbul,_ Turkey 3Department of Civil Engineering, Faculty of Engineering and Architecture, Geli¸simUniversity, Istanbul,_ Turkey 4Department of Mathematics, Faculty of Science and Arts, Uluda˘gUniversity, Bursa, Turkey Received: 13.04.2016 • Accepted/Published Online: 14.02.2017 • Final Version: 23.11.2017 Abstract: In this paper, by considering a Frenet curve lying on an oriented hypersurface, we extend the Darboux frame field into Euclidean 4-space E4 . Depending on the linear independency of the curvature vector with the hypersurface's normal, we obtain two cases for this extension. For each case, we obtain some geometrical meanings of new invariants along the curve on the hypersurface. We also give the relationships between the Frenet frame curvatures and Darboux frame curvatures in E4 . Finally, we compute the expressions of the new invariants of a Frenet curve lying on an implicit hypersurface. Key words: Curves on hypersurface, Darboux frame field, curvatures 1. Introduction In differential geometry, frame fields constitute an important tool while studying curves and surfaces. The most familiar frame fields are the Frenet{Serret frame along a space curve, and the Darboux frame along a surface curve.
  • Two Exotic Holonomies in Dimension Four, Path Geometries, and Twistor Theory

    Two Exotic Holonomies in Dimension Four, Path Geometries, and Twistor Theory

    Two Exotic Holonomies in Dimension Four, Path Geometries, and Twistor Theory by Robert L. Bryant* Table of Contents §0. Introduction §1. The Holonomy of a Torsion-Free Connection §2. The Structure Equations of H3-andG3-structures §3. The Existence Theorems §4. Path Geometries and Exotic Holonomy §5. Twistor Theory and Exotic Holonomy §6. Epilogue §0. Introduction Since its introduction by Elie´ Cartan, the holonomy of a connection has played an important role in differential geometry. One of the best known results concerning hol- onomy is Berger’s classification of the possible holonomies of Levi-Civita connections of Riemannian metrics. Since the appearance of Berger [1955], much work has been done to refine his listofpossible Riemannian holonomies. See the recent works by Besse [1987]or Salamon [1989]foruseful historical surveys and applications of holonomy to Riemannian and algebraic geometry. Less well known is that, at the same time, Berger also classified the possible pseudo- Riemannian holonomies which act irreducibly on each tangent space. The intervening years have not completely resolved the question of whether all of the subgroups of the general linear group on his pseudo-Riemannian list actually occur as holonomy groups of pseudo-Riemannian metrics, but, as of this writing, only one class of examples on his list remains in doubt, namely SO∗(2p) ⊂ GL(4p)forp ≥ 3. Perhaps the least-known aspect of Berger’s work on holonomy concerns his list of the possible irreducibly-acting holonomy groups of torsion-free affine connections. (Torsion- free connections are the natural generalization of Levi-Civita connections in the metric case.) Part of the reason for this relative obscurity is that affine geometry is not as well known or widely usedasRiemannian geometry and global results are generally lacking.
  • J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2019 This Course Is an Introduction to the Geometry of Smooth Curves and Surf

    J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2019 This Course Is an Introduction to the Geometry of Smooth Curves and Surf

    J.M. Sullivan, TU Berlin A: Curves Diff Geom I, SS 2019 This course is an introduction to the geometry of smooth if the velocity never vanishes). Then the speed is a (smooth) curves and surfaces in Euclidean space Rn (in particular for positive function of t. (The cusped curve β above is not regular n = 2; 3). The local shape of a curve or surface is described at t = 0; the other examples given are regular.) in terms of its curvatures. Many of the big theorems in the DE The lengthR [ : Länge] of a smooth curve α is defined as subject – such as the Gauss–Bonnet theorem, a highlight at the j j len(α) = I α˙(t) dt. (For a closed curve, of course, we should end of the semester – deal with integrals of curvature. Some integrate from 0 to T instead of over the whole real line.) For of these integrals are topological constants, unchanged under any subinterval [a; b] ⊂ I, we see that deformation of the original curve or surface. Z b Z b We will usually describe particular curves and surfaces jα˙(t)j dt ≥ α˙(t) dt = α(b) − α(a) : locally via parametrizations, rather than, say, as level sets. a a Whereas in algebraic geometry, the unit circle is typically be described as the level set x2 + y2 = 1, we might instead This simply means that the length of any curve is at least the parametrize it as (cos t; sin t). straight-line distance between its endpoints. Of course, by Euclidean space [DE: euklidischer Raum] The length of an arbitrary curve can be defined (following n we mean the vector space R 3 x = (x1;:::; xn), equipped Jordan) as its total variation: with with the standard inner product or scalar product [DE: P Xn Skalarproduktp ] ha; bi = a · b := aibi and its associated norm len(α):= TV(α):= sup α(ti) − α(ti−1) : jaj := ha; ai.
  • Curvatures of Smooth and Discrete Surfaces

    Curvatures of Smooth and Discrete Surfaces

    Curvatures of Smooth and Discrete Surfaces John M. Sullivan The curvatures of a smooth curve or surface are local measures of its shape. Here we consider analogous quantities for discrete curves and surfaces, meaning polygonal curves and triangulated polyhedral surfaces. We find that the most useful analogs are those which preserve integral relations for curvature, like the Gauß/Bonnet theorem. For simplicity, we usually restrict our attention to curves and surfaces in euclidean three space E3, although many of the results would easily generalize to other ambient manifolds of arbitrary dimension. Most of the material here is not new; some is even quite old. Although some refer- ences are given, no attempt has been made to give a comprehensive bibliography or an accurate picture of the history of the ideas. 1. Smooth curves, Framings and Integral Curvature Relations A companion article [Sul06] investigated the class of FTC (finite total curvature) curves, which includes both smooth and polygonal curves, as a way of unifying the treatment of curvature. Here we briefly review the theory of smooth curves from the point of view we 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. The only first-order information is the tangent line; since all lines in space are equivalent, there are no first-order invariants. Second-order information is given by the osculating circle, and the invariant is its curvature κ = 1/r. For a plane curve given as a graph y = f(x) let us contrast the notions of curvature and second derivative.
  • Differential Geometry

    Differential Geometry

    Differential Geometry J.B. Cooper 1995 Inhaltsverzeichnis 1 CURVES AND SURFACES—INFORMAL DISCUSSION 2 1.1 Surfaces ................................ 13 2 CURVES IN THE PLANE 16 3 CURVES IN SPACE 29 4 CONSTRUCTION OF CURVES 35 5 SURFACES IN SPACE 41 6 DIFFERENTIABLEMANIFOLDS 59 6.1 Riemannmanifolds .......................... 69 1 1 CURVES AND SURFACES—INFORMAL DISCUSSION We begin with an informal discussion of curves and surfaces, concentrating on methods of describing them. We shall illustrate these with examples of classical curves and surfaces which, we hope, will give more content to the material of the following chapters. In these, we will bring a more rigorous approach. Curves in R2 are usually specified in one of two ways, the direct or parametric representation and the implicit representation. For example, straight lines have a direct representation as tx + (1 t)y : t R { − ∈ } i.e. as the range of the function φ : t tx + (1 t)y → − (here x and y are distinct points on the line) and an implicit representation: (ξ ,ξ ): aξ + bξ + c =0 { 1 2 1 2 } (where a2 + b2 = 0) as the zero set of the function f(ξ ,ξ )= aξ + bξ c. 1 2 1 2 − Similarly, the unit circle has a direct representation (cos t, sin t): t [0, 2π[ { ∈ } as the range of the function t (cos t, sin t) and an implicit representation x : 2 2 → 2 2 { ξ1 + ξ2 =1 as the set of zeros of the function f(x)= ξ1 + ξ2 1. We see from} these examples that the direct representation− displays the curve as the image of a suitable function from R (or a subset thereof, usually an in- terval) into two dimensional space, R2.
  • Lines of Curvature on Surfaces, Historical Comments and Recent Developments

    Lines of Curvature on Surfaces, Historical Comments and Recent Developments

    S˜ao Paulo Journal of Mathematical Sciences 2, 1 (2008), 99–143 Lines of Curvature on Surfaces, Historical Comments and Recent Developments Jorge Sotomayor Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Rua do Mat˜ao 1010, Cidade Universit´aria, CEP 05508-090, S˜ao Paulo, S.P., Brazil Ronaldo Garcia Instituto de Matem´atica e Estat´ıstica, Universidade Federal de Goi´as, CEP 74001-970, Caixa Postal 131, Goiˆania, GO, Brazil Abstract. This survey starts with the historical landmarks leading to the study of principal configurations on surfaces, their structural sta- bility and further generalizations. Here it is pointed out that in the work of Monge, 1796, are found elements of the qualitative theory of differential equations ( QTDE ), founded by Poincar´ein 1881. Here are also outlined a number of recent results developed after the assimilation into the subject of concepts and problems from the QTDE and Dynam- ical Systems, such as Structural Stability, Bifurcations and Genericity, among others, as well as extensions to higher dimensions. References to original works are given and open problems are proposed at the end of some sections. 1. Introduction The book on differential geometry of D. Struik [79], remarkable for its historical notes, contains key references to the classical works on principal curvature lines and their umbilic singularities due to L. Euler [8], G. Monge [61], C. Dupin [7], G. Darboux [6] and A. Gullstrand [39], among others (see The authors are fellows of CNPq and done this work under the project CNPq 473747/2006-5. The authors are grateful to L.
  • On the Patterns of Principal Curvature Lines Around a Curve of Umbilic Points

    On the Patterns of Principal Curvature Lines Around a Curve of Umbilic Points

    Anais da Academia Brasileira de Ciências (2005) 77(1): 13–24 (Annals of the Brazilian Academy of Sciences) ISSN 0001-3765 www.scielo.br/aabc On the Patterns of Principal Curvature Lines around a Curve of Umbilic Points RONALDO GARCIA1 and JORGE SOTOMAYOR2 1Instituto de Matemática e Estatística, Universidade Federal de Goiás Caixa Postal 131 – 74001-970 Goiânia, GO, Brasil 2Instituto de Matemática e Estatística, Universidade de São Paulo Rua do Matão 1010, Cidade Universitária, 05508-090 São Paulo, SP, Brasil Manuscript received on June 15, 2004; accepted for publication on October 10, 2004; contributed by Jorge Sotomayor* ABSTRACT In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface. Key words: Umbilic point, principal curvature lines, principal cycles. 1 INTRODUCTION The study of umbilic points on surfaces and the patterns of principal curvature lines around them has attracted the attention of generation of mathematicians among whom can be named Monge, Darboux and Carathéodory. One aspect – concerning isolated umbilics – of the contributions of these authors, departing from Darboux (Darboux 1896), has been elaborated and extended in several directions by Garcia, Sotomayor and Gutierrez, among others. See (Gutierrez and Sotomayor, 1982, 1991, 1998), (Garcia and Sotomayor, 1997, 2000) and (Garcia et al. 2000, 2004) where additional references can be found. In (Carathéodory 1935) Carathéodory mentioned the interest of non isolated umbilics in generic surfaces pertinent to Geometric Optics. In a remarkably concise study he established that any local analytic regular arc of curve in R3 is a curve of umbilic points of a piece of analytic surface.