DOCSLIB.ORG

Riemann Surfaces by Way of Complex Analytic Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Riemann Surfaces by Way of Complex Analytic Geometry Riemann Surfaces by Way of Complex Analytic Geometry Dror Varolin Graduate Studies in Mathematics Volume 125 American Mathematical Society http://dx.doi.org/10.1090/gsm/125 Riemann Surfaces by Way of Complex Analytic Geometry Riemann Surfaces by Way of Complex Analytic Geometry Dror Varolin Graduate Studies in Mathematics Volume 125 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Rafe Mazzeo Martin Scharlemann Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 30F10, 30F15, 30F30, 30F45, 30F99, 30G99, 31A05, 31A99, 32W05. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-125 Library of Congress Cataloging-in-Publication Data Varolin, Dror, 1970– Riemann surfaces by way of complex analytic geometry / Dror Varolin. p. cm. — (Graduate studies in mathematics ; v. 125) Includes bibliographical references and index. ISBN 978-0-8218-5369-6 (alk. paper) 1. Riemann surfaces. 2. Functions of complex variables. 3. Geometry, Analytic. I. Title. II. Series. QA333.V37 2011 515.93—dc23 2011014621 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2011 by Dror Varolin. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 Dedicated to Erin Contents Preface xi Chapter 1. Complex Analysis 1 §1.1. Green’s Theorem and the Cauchy-Green Formula 1 §1.2. Holomorphic functions and Cauchy Formulas 2 §1.3. Power series 3 §1.4. Isolated singularities of holomorphic functions 4 §1.5. The Maximum Principle 8 §1.6. Compactness theorems 9 §1.7. Harmonic functions 11 §1.8. Subharmonic functions 14 §1.9. Exercises 19 Chapter 2. Riemann Surfaces 21 §2.1. Definition of a Riemann surface 21 §2.2. Riemann surfaces as smooth 2-manifolds 23 §2.3. Examples of Riemann surfaces 25 §2.4. Exercises 36 Chapter 3. Functions on Riemann Surfaces 37 §3.1. Holomorphic and meromorphic functions 37 §3.2. Global aspects of meromorphic functions 42 §3.3. Holomorphic maps between Riemann surfaces 45 §3.4. An example: Hyperelliptic surfaces 54 vii viii Contents §3.5. Harmonic and subharmonic functions 57 §3.6. Exercises 59 Chapter 4. Complex Line Bundles 61 §4.1. Complex line bundles 61 §4.2. Holomorphic line bundles 65 §4.3. Two canonically defined holomorphic line bundles 66 §4.4. Holomorphic vector fields on a Riemann surface 70 §4.5. Divisors and line bundles 74 §4.6. Line bundles over Pn 79 §4.7. Holomorphic sections and projective maps 81 §4.8. A finiteness theorem 84 §4.9. Exercises 85 Chapter 5. Complex Differential Forms 87 §5.1. Differential (1, 0)-forms 87 § ∗0,1 5.2. TX and (0, 1)-forms 89 § ∗ 5.3. TX and 1-forms 89 § 1,1 5.4. ΛX and (1, 1)-forms 90 §5.5. Exterior algebra and calculus 90 §5.6. Integration of forms 92 §5.7. Residues 95 §5.8. Homotopy and homology 96 §5.9. Poincare´ and Dolbeault Lemmas 98 §5.10. Dolbeault cohomology 99 §5.11. Exercises 100 Chapter 6. Calculus on Line Bundles 101 §6.1. Connections on line bundles 101 §6.2. Hermitian metrics and connections 104 §6.3. (1, 0)-connections on holomorphic line bundles 105 §6.4. The Chern connection 106 §6.5. Curvature of the Chern connection 107 §6.6. Chern numbers 109 § 1,0 6.7. Example: The holomorphic line bundle TX 111 §6.8. Exercises 112 Chapter 7. Potential Theory 115 Contents ix §7.1. The Dirichlet Problem and Perron’s Method 115 §7.2. Approximation on open Riemann surfaces 126 §7.3. Exercises 130 Chapter 8. Solving ∂¯ for Smooth Data 133 §8.1. The basic result 133 §8.2. Triviality of holomorphic line bundles 134 §8.3. The Weierstrass Product Theorem 135 §8.4. Meromorphic functions as quotients 135 §8.5. The Mittag-Leffler Problem 136 §8.6. The Poisson Equation on open Riemann surfaces 140 §8.7. Exercises 143 Chapter 9. Harmonic Forms 145 §9.1. The definition and basic properties of harmonic forms 145 §9.2. Harmonic forms and cohomology 149 §9.3. The Hodge decomposition of E (X) 151 §9.4. Existence of positive line bundles 157 §9.5. Proof of the Dolbeault-Serre isomorphism 161 §9.6. Exercises 161 Chapter 10. Uniformization 165 §10.1. Automorphisms of the complex plane, projective line, and unit disk 165 §10.2. A review of covering spaces 166 §10.3. The Uniformization Theorem 168 §10.4. Proof of the Uniformization Theorem 174 §10.5. Exercises 175 Chapter 11. Hormander’s¨ Theorem 177 §11.1. Hilbert spaces of sections 177 §11.2. The Basic Identity 180 §11.3. Hormander’s¨ Theorem 183 §11.4. Proof of the Korn-Lichtenstein Theorem 184 §11.5. Exercises 195 Chapter 12. Embedding Riemann Surfaces 197 §12.1. Controlling the derivatives of sections 198 §12.2. Meromorphic sections of line bundles 201 x Contents §12.3. Plenitude of meromorphic functions 202 §12.4. Kodaira’s Embedding Theorem 202 §12.5. Narasimhan’s Embedding Theorem 204 §12.6. Exercises 210 Chapter 13. The Riemann-Roch Theorem 211 §13.1. The Riemann-Roch Theorem 211 §13.2. Some corollaries 217 Chapter 14. Abel’s Theorem 223 §14.1. Indefinite integration of holomorphic forms 223 §14.2. Riemann’s Bilinear Relations 225 §14.3. The Reciprocity Theorem 228 §14.4. Proof of Abel’s Theorem 229 §14.5. A discussion of Jacobi’s Inversion Theorem 231 Bibliography 233 Index 235 Preface What This book presents certain parts of the basic theory of Riemann surfaces through methods of complex analytic geometry, many of which were developed at one time or another in the past 50 years or so. The first chapter of the book presents a rapid review of the elementary part of basic complex analysis, introducing holomorphic, meromorphic, harmonic, and subharmonic functions and establishing some of the basic local properties of these functions. In the second chapter we define Riemann surfaces and give some exam- ples, both of a concrete and abstract nature. The most abstract example is that any oriented 2-dimensional real manifold has a complex structure. In fact, we state the classical theorem of Korn-Lichtenstein: given any metric on an oriented surface, there is a complex structure that makes that metric locally conformal to the Eu- clidean metric in any holomorphic chart. The proof of this theorem is given only in Chapter 11. Riemann surfaces are non-linear spaces on which, roughly speaking, local complex analysis makes sense, and so we can extend the inherently local notions of holomorphic, meromorphic, harmonic, and subharmonic functions to such sur- faces. This point of view starts the third chapter, but by the end we prove our first main result, namely the constancy of the degree of a holomorphic map between two compact Riemann surfaces. With the recollection of some basic topology of surfaces, we are then able to establish the Riemann-Hurwitz Formula, which puts constraints on holomorphic maps between compact Riemann surfaces. The chap- ter ends with a proof of the Harnack Principle. Up to this point, the presentation is mostly classical, in the sense that it does not differ markedly from other presen- tations of the same material (though perhaps the particular blend of the ideas is unusual). xi xii Preface The novelty of the book begins in the fourth chapter, where we first define and give examples of complex and holomorphic line bundles. There are a number of important, albeit elementary, ideas that are presented in this chapter, and among these the correspondence between divisors and holomorphic line bundles with sec- tions is first established. Another key idea is the correspondence between finite- dimensional vector spaces of sections of a holomorphic line bundle and projective maps. The chapter ends with an elementary proof of the finite-dimensionality of the space of all global holomorphic sections of a holomorphic line bundle over a compact Riemann surface. The fifth chapter begins by defining complex differential forms. These forms play a central role in the rest of the book, but in this particular chapter we focus on their formal aspects—how to integrate them and how to describe the topology of a Riemann surface with them. In the sixth chapter we develop the theory of differentiation for sections of complex line bundles (which is known as defining a connection). A complex line bundle admits many connections and, unlike the trivial bundle, there is no canon- ical choice of a connection for a general complex line bundle. The first goal of the chapter is to describe geometric structures that isolate a subset of the possible ways to differentiate sections. If a geometry is symmetric enough, there will be a canonical way to define differentiation of sections. The first instance of such a scenario is the classical Levi-Civitaˇ Theorem for vector fields on a Riemannian manifold. The main fact we establish in Chapter 6, due to Chern, is that for each holomorphic line bundle with Hermitian metric there is a unique connection that is compatible with these structures in the appropriate sense.
Load more

Recommended publications
  • Chapter 11. Three Dimensional Analytic Geometry and Vectors
    Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1.
    [Show full text]
  • 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]
  • Section 2.6 Cylindrical and Spherical Coordinates
    Section 2.6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O,the rotating ray or half line from O with unit tick. A point P in the plane can be uniquely described by its distance to the origin r = dist (P, O) and the angle µ, 0 µ < 2¼ : · Y P(x,y) r θ O X We call (r, µ) the polar coordinate of P. Suppose that P has Cartesian (stan- dard rectangular) coordinate (x, y) .Then the relation between two coordinate systems is displayed through the following conversion formula: x = r cos µ Polar Coord. to Cartesian Coord.: y = r sin µ ½ r = x2 + y2 Cartesian Coord. to Polar Coord.: y tan µ = ( p x 0 µ < ¼ if y > 0, 2¼ µ < ¼ if y 0. · · · Note that function tan µ has period ¼, and the principal value for inverse tangent function is ¼ y ¼ < arctan < . ¡ 2 x 2 1 So the angle should be determined by y arctan , if x > 0 xy 8 arctan + ¼, if x < 0 µ = > ¼ x > > , if x = 0, y > 0 < 2 ¼ , if x = 0, y < 0 > ¡ 2 > > Example 6.1. Fin:>d (a) Cartesian Coord. of P whose Polar Coord. is ¼ 2, , and (b) Polar Coord. of Q whose Cartesian Coord. is ( 1, 1) . 3 ¡ ¡ ³ So´l. (a) ¼ x = 2 cos = 1, 3 ¼ y = 2 sin = p3. 3 (b) r = p1 + 1 = p2 1 ¼ ¼ 5¼ tan µ = ¡ = 1 = µ = or µ = + ¼ = . 1 ) 4 4 4 ¡ 5¼ Since ( 1, 1) is in the third quadrant, we choose µ = so ¡ ¡ 4 5¼ p2, is Polar Coord.
    [Show full text]
  • Area, Volume and Surface Area
    The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 YEARS 810 Area, Volume and Surface Area (Measurement and Geometry: Module 11) For teachers of Primary and Secondary Mathematics 510 Cover design, Layout design and Typesetting by Claire Ho The Improving Mathematics Education in Schools (TIMES) Project 2009‑2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations. © The University of Melbourne on behalf of the international Centre of Excellence for Education in Mathematics (ICE‑EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution‑NonCommercial‑NoDerivs 3.0 Unported License. http://creativecommons.org/licenses/by‑nc‑nd/3.0/ The Improving Mathematics Education in Schools (TIMES) Project MEASUREMENT AND GEOMETRY Module 11 AREA, VOLUME AND SURFACE AREA A guide for teachers - Years 8–10 June 2011 Peter Brown Michael Evans David Hunt Janine McIntosh Bill Pender Jacqui Ramagge YEARS 810 {4} A guide for teachers AREA, VOLUME AND SURFACE AREA ASSUMED KNOWLEDGE • Knowledge of the areas of rectangles, triangles, circles and composite figures. • The definitions of a parallelogram and a rhombus. • Familiarity with the basic properties of parallel lines. • Familiarity with the volume of a rectangular prism. • Basic knowledge of congruence and similarity. • Since some formulas will be involved, the students will need some experience with substitution and also with the distributive law.
    [Show full text]
  • Projective Geometry: a Short Introduction
    Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g.
    [Show full text]
  • Algebraic Geometry Michael Stoll
    Introductory Geometry Course No. 100 351 Fall 2005 Second Part: Algebraic Geometry Michael Stoll Contents 1. What Is Algebraic Geometry? 2 2. Affine Spaces and Algebraic Sets 3 3. Projective Spaces and Algebraic Sets 6 4. Projective Closure and Affine Patches 9 5. Morphisms and Rational Maps 11 6. Curves — Local Properties 14 7. B´ezout’sTheorem 18 2 1. What Is Algebraic Geometry? Linear Algebra can be seen (in parts at least) as the study of systems of linear equations. In geometric terms, this can be interpreted as the study of linear (or affine) subspaces of Cn (say). Algebraic Geometry generalizes this in a natural way be looking at systems of polynomial equations. Their geometric realizations (their solution sets in Cn, say) are called algebraic varieties. Many questions one can study in various parts of mathematics lead in a natural way to (systems of) polynomial equations, to which the methods of Algebraic Geometry can be applied. Algebraic Geometry provides a translation between algebra (solutions of equations) and geometry (points on algebraic varieties). The methods are mostly algebraic, but the geometry provides the intuition. Compared to Differential Geometry, in Algebraic Geometry we consider a rather restricted class of “manifolds” — those given by polynomial equations (we can allow “singularities”, however). For example, y = cos x defines a perfectly nice differentiable curve in the plane, but not an algebraic curve. In return, we can get stronger results, for example a criterion for the existence of solutions (in the complex numbers), or statements on the number of solutions (for example when intersecting two curves), or classification results.
    [Show full text]
  • Arxiv:1910.11630V1 [Math.AG] 25 Oct 2019 3 Geometric Invariant Theory 10 3.1 Quotients and the Notion of Stability
    Geometric Invariant Theory, holomorphic vector bundles and the Harder–Narasimhan filtration Alfonso Zamora Departamento de Matem´aticaAplicada y Estad´ıstica Universidad CEU San Pablo Juli´anRomea 23, 28003 Madrid, Spain e-mail: [email protected] Ronald A. Z´u˜niga-Rojas Centro de Investigaciones Matem´aticasy Metamatem´aticas CIMM Escuela de Matem´atica,Universidad de Costa Rica UCR San Jos´e11501, Costa Rica e-mail: [email protected] Abstract. This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Narasimhan filtration and, from which, correspondences with the GIT picture and results derived from stratifications on the moduli space are discussed. Keywords: Geometric Invariant Theory, Harder-Narasimhan filtration, moduli spaces, vector bundles, Higgs bundles, GIT stability, symplectic stability, stratifications. MSC class: 14D07, 14D20, 14H10, 14H60, 53D30 Contents 1 Introduction 2 2 Preliminaries 4 2.1 Lie groups . .4 2.2 Lie algebras . .6 2.3 Algebraic varieties . .7 2.4 Vector bundles . .8 arXiv:1910.11630v1 [math.AG] 25 Oct 2019 3 Geometric Invariant Theory 10 3.1 Quotients and the notion of stability . 10 3.2 Hilbert-Mumford criterion . 14 3.3 Symplectic stability . 18 3.4 Examples . 21 3.5 Maximal unstability . 24 2 git, hvb & hnf 4 Moduli Space of vector bundles 28 4.1 GIT construction of the moduli space . 28 4.2 Harder-Narasimhan filtration .
    [Show full text]
  • Analytic Geometry
    STATISTIC ANALYTIC GEOMETRY SESSION 3 STATISTIC SESSION 3 Session 3 Analytic Geometry Geometry is all about shapes and their properties. If you like playing with objects, or like drawing, then geometry is for you! Geometry can be divided into: Plane Geometry is about flat shapes like lines, circles and triangles ... shapes that can be drawn on a piece of paper Solid Geometry is about three dimensional objects like cubes, prisms, cylinders and spheres Point, Line, Plane and Solid A Point has no dimensions, only position A Line is one-dimensional A Plane is two dimensional (2D) A Solid is three-dimensional (3D) Plane Geometry Plane Geometry is all about shapes on a flat surface (like on an endless piece of paper). 2D Shapes Activity: Sorting Shapes Triangles Right Angled Triangles Interactive Triangles Quadrilaterals (Rhombus, Parallelogram, etc) Rectangle, Rhombus, Square, Parallelogram, Trapezoid and Kite Interactive Quadrilaterals Shapes Freeplay Perimeter Area Area of Plane Shapes Area Calculation Tool Area of Polygon by Drawing Activity: Garden Area General Drawing Tool Polygons A Polygon is a 2-dimensional shape made of straight lines. Triangles and Rectangles are polygons. Here are some more: Pentagon Pentagra m Hexagon Properties of Regular Polygons Diagonals of Polygons Interactive Polygons The Circle Circle Pi Circle Sector and Segment Circle Area by Sectors Annulus Activity: Dropping a Coin onto a Grid Circle Theorems (Advanced Topic) Symbols There are many special symbols used in Geometry. Here is a short reference for you:
    [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]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Vector Bundles and Projective Modules
    VECTOR BUNDLES AND PROJECTIVE MODULES BY RICHARD G. SWAN(i) Serre [9, §50] has shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective mo- dules over its coordinate ring. For some time, it has been assumed that a similar correspondence exists between topological vector bundles over a compact Haus- dorff space X and finitely generated projective modules over the ring of con- tinuous real-valued functions on X. A number of examples of projective modules have been given using this correspondence. However, no rigorous treatment of the correspondence seems to have been given. I will give such a treatment here and then give some of the examples which may be constructed in this way. 1. Preliminaries. Let K denote either the real numbers, complex numbers or quaternions. A X-vector bundle £ over a topological space X consists of a space F(£) (the total space), a continuous map p : E(Ç) -+ X (the projection) which is onto, and, on each fiber Fx(¡z)= p-1(x), the structure of a finite di- mensional vector space over K. These objects are required to satisfy the follow- ing condition: for each xeX, there is a neighborhood U of x, an integer n, and a homeomorphism <p:p-1(U)-> U x K" such that on each fiber <b is a X-homo- morphism. The fibers u x Kn of U x K" are X-vector spaces in the obvious way. Note that I do not require n to be a constant.
    [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]