Compact Riemann Surfaces
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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. -
Equivelar Octahedron of Genus 3 in 3-Space
Equivelar octahedron of genus 3 in 3-space Ruslan Mizhaev ([email protected]) Apr. 2020 ABSTRACT . Building up a toroidal polyhedron of genus 3, consisting of 8 nine-sided faces, is given. From the point of view of topology, a polyhedron can be considered as an embedding of a cubic graph with 24 vertices and 36 edges in a surface of genus 3. This polyhedron is a contender for the maximal genus among octahedrons in 3-space. 1. Introduction This solution can be attributed to the problem of determining the maximal genus of polyhedra with the number of faces - . As is known, at least 7 faces are required for a polyhedron of genus = 1 . For cases ≥ 8 , there are currently few examples. If all faces of the toroidal polyhedron are – gons and all vertices are q-valence ( ≥ 3), such polyhedral are called either locally regular or simply equivelar [1]. The characteristics of polyhedra are abbreviated as , ; [1]. 2. Polyhedron {9, 3; 3} V1. The paper considers building up a polyhedron 9, 3; 3 in 3-space. The faces of a polyhedron are non- convex flat 9-gons without self-intersections. The polyhedron is symmetric when rotated through 1804 around the axis (Fig. 3). One of the features of this polyhedron is that any face has two pairs with which it borders two edges. The polyhedron also has a ratio of angles and faces - = − 1. To describe polyhedra with similar characteristics ( = − 1) we use the Euler formula − + = = 2 − 2, where is the Euler characteristic. Since = 3, the equality 3 = 2 holds true. -
LOOKING BACKWARD: from EULER to RIEMANN 11 at the Age of 24
LOOKING BACKWARD: FROM EULER TO RIEMANN ATHANASE PAPADOPOULOS Il est des hommes auxquels on ne doit pas adresser d’´eloges, si l’on ne suppose pas qu’ils ont le goˆut assez peu d´elicat pour goˆuter les louanges qui viennent d’en bas. (Jules Tannery, [241] p. 102) Abstract. We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and the notion of space. We shall see that among Riemann’s predecessors in all these fields, one name occupies a prominent place, this is Leonhard Euler. The final version of this paper will appear in the book From Riemann to differential geometry and relativity (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017. AMS Mathematics Subject Classification: 01-02, 01A55, 01A67, 26A42, 30- 03, 33C05, 00A30. Keywords: Bernhard Riemann, function of a complex variable, space, Rie- mannian geometry, trigonometric series, zeta function, differential geometry, elliptic integral, elliptic function, Abelian integral, Abelian function, hy- pergeometric function, topology, Riemann surface, Leonhard Euler, space, integration. Contents 1. Introduction 2 2. Functions 10 3. Elliptic integrals 21 arXiv:1710.03982v1 [math.HO] 11 Oct 2017 4. Abelian functions 30 5. Hypergeometric series 32 6. The zeta function 34 7. On space 41 8. Topology 47 9. Differential geometry 63 10. Trigonometric series 69 11. Integration 79 12. Conclusion 81 References 85 Date: October 12, 2017. -
Dynamical Systems/Syst`Emes Dynamiques SIMPLE
Dynamical systems/Systemes` dynamiques SIMPLE EXPONENTIAL ESTIMATE FOR THE NUMBER OF REAL ZEROS OF COMPLETE ABELIAN INTEGRALS Dmitri Novikov, Sergeı˘ Yakovenko The Weizmann Institute of Science (Rehovot, Israel) Abstract. We show that for a generic real polynomial H = H(x, y) and an arbitrary real differential 1-form ω = P (x, y) dx + Q(x, y) dy with polynomial coefficients of degree 6 d, the number of ovals of the foliation H = const, which yield the zero value of the complete H Abelian integral I(t) = H=t ω, grows at most as exp OH (d) as d → ∞, where OH (d) depends only on H. This result essentially improves the previous double exponential estimate obtained in [1] for the number of zero ovals on a bounded distance from singular level curves of H. La borne simplement exponentielle pour le nombre de zeros´ reels´ isoles´ des integrales completes` abeliennes´ R´esum´e. Soit H = H(x, y) un polynˆome r´eel de deux variables et ω = P (x, y) dx + Q(x, y) dy une forme diff´erentielle quelconque avec des coefficients polynomiaux r´eels de degr´e d. Nous montrons que le nombre des ovales (c’est-a-dire les composantes compactes connexes) des courbes de niveau H = const, telles que l’int´egrale de la forme s’annule, est au plus exp OH (d) quand d → ∞, o`u OH (d) ne d´epend que du polynˆome H. Ce th´eor`emeam´eliore de mani`ere essentielle la borne doublement exponentielle obtenue dans [1]. Nous ´etablissons aussi les bornes sup´erieures exponentielles similaires pour le nom- bre de z´eros isol´espour les enveloppes polynomiales des ´equations diff´erentielles ordinaires lin´eaires irr´eductibles ou presque irr´eductibles. -
Lectures on Modular Forms. Fall 1997/98
Lectures on Modular Forms. Fall 1997/98 Igor V. Dolgachev October 26, 2017 ii Contents 1 Binary Quadratic Forms1 2 Complex Tori 13 3 Theta Functions 25 4 Theta Constants 43 5 Transformations of Theta Functions 53 6 Modular Forms 63 7 The Algebra of Modular Forms 83 8 The Modular Curve 97 9 Absolute Invariant and Cross-Ratio 115 10 The Modular Equation 121 11 Hecke Operators 133 12 Dirichlet Series 147 13 The Shimura-Tanyama-Weil Conjecture 159 iii iv CONTENTS Lecture 1 Binary Quadratic Forms 1.1 The theory of modular form originates from the work of Carl Friedrich Gauss of 1831 in which he gave a geometrical interpretation of some basic no- tions of number theory. Let us start with choosing two non-proportional vectors v = (v1; v2) and w = 2 (w1; w2) in R The set of vectors 2 Λ = Zv + Zw := fm1v + m2w 2 R j m1; m2 2 Zg forms a lattice in R2, i.e., a free subgroup of rank 2 of the additive group of the vector space R2. We picture it as follows: • • • • • • •Gv • ••• •• • w • • • • • • • • Figure 1.1: Lattice in R2 1 2 LECTURE 1. BINARY QUADRATIC FORMS Let v v B(v; w) = 1 2 w1 w2 and v · v v · w G(v; w) = = B(v; w) · tB(v; w): v · w w · w be the Gram matrix of (v; w). The area A(v; w) of the parallelogram formed by the vectors v and w is given by the formula v · v v · w A(v; w)2 = det G(v; w) = (det B(v; w))2 = det : v · w w · w Let x = mv + nw 2 Λ. -
Riemann Surfaces
RIEMANN SURFACES AARON LANDESMAN CONTENTS 1. Introduction 2 2. Maps of Riemann Surfaces 4 2.1. Defining the maps 4 2.2. The multiplicity of a map 4 2.3. Ramification Loci of maps 6 2.4. Applications 6 3. Properness 9 3.1. Definition of properness 9 3.2. Basic properties of proper morphisms 9 3.3. Constancy of degree of a map 10 4. Examples of Proper Maps of Riemann Surfaces 13 5. Riemann-Hurwitz 15 5.1. Statement of Riemann-Hurwitz 15 5.2. Applications 15 6. Automorphisms of Riemann Surfaces of genus ≥ 2 18 6.1. Statement of the bound 18 6.2. Proving the bound 18 6.3. We rule out g(Y) > 1 20 6.4. We rule out g(Y) = 1 20 6.5. We rule out g(Y) = 0, n ≥ 5 20 6.6. We rule out g(Y) = 0, n = 4 20 6.7. We rule out g(C0) = 0, n = 3 20 6.8. 21 7. Automorphisms in low genus 0 and 1 22 7.1. Genus 0 22 7.2. Genus 1 22 7.3. Example in Genus 3 23 Appendix A. Proof of Riemann Hurwitz 25 Appendix B. Quotients of Riemann surfaces by automorphisms 29 References 31 1 2 AARON LANDESMAN 1. INTRODUCTION In this course, we’ll discuss the theory of Riemann surfaces. Rie- mann surfaces are a beautiful breeding ground for ideas from many areas of math. In this way they connect seemingly disjoint fields, and also allow one to use tools from different areas of math to study them. -
Abelian Varieties and Theta Functions Associated to Compact Riemannian Manifolds; Constructions Inspired by Superstring Theory
ABELIAN VARIETIES AND THETA FUNCTIONS ASSOCIATED TO COMPACT RIEMANNIAN MANIFOLDS; CONSTRUCTIONS INSPIRED BY SUPERSTRING THEORY. S. MULLER-STACH,¨ C. PETERS AND V. SRINIVAS MATH. INST., JOHANNES GUTENBERG UNIVERSITAT¨ MAINZ, INSTITUT FOURIER, UNIVERSITE´ GRENOBLE I ST.-MARTIN D'HERES,` FRANCE AND TIFR, MUMBAI, INDIA Resum´ e.´ On d´etailleune construction d^ue Witten et Moore-Witten (qui date d'environ 2000) d'une vari´et´eab´elienneprincipalement pola- ris´eeassoci´ee`aune vari´et´ede spin. Le th´eor`emed'indice pour l'op´erateur de Dirac (associ´e`ala structure de spin) implique qu'un accouplement naturel sur le K-groupe topologique prend des valeurs enti`eres.Cet ac- couplement sert commme polarization principale sur le t^oreassoci´e. On place la construction dans un c^adreg´en´eralce qui la relie `ala ja- cobienne de Weil mais qui sugg`ereaussi la construction d'une jacobienne associ´ee`an'importe quelle structure de Hodge polaris´eeet de poids pair. Cette derni`ereconstruction est ensuite expliqu´eeen termes de groupes alg´ebriques,utile pour le point de vue des cat´egoriesTannakiennes. Notre construction depend de param`etres,beaucoup comme dans la th´eoriede Teichm¨uller,mais en g´en´erall'application de p´eriodes n'est que de nature analytique r´eelle. Abstract. We first investigate a construction of principally polarized abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000. The index theorem for the Dirac operator associated to the spin structure implies integrality of a natural skew pairing on the topological K-group. The latter serves as a principal polarization. -
Weierstrass Points of Superelliptic Curves
Weierstrass points of superelliptic curves C. SHOR a T. SHASKA b a Department of Mathematics, Western New England University, Springfield, MA, USA; E-mail: [email protected] b Department of Mathematics, Oakland University, Rochester, MI, USA; E-mail: [email protected] Abstract. In this lecture we give a brief introduction to Weierstrass points of curves and computational aspects of q-Weierstrass points on superel- liptic curves. Keywords. hyperelliptic curves, superelliptic curves, Weierstrass points Introduction These lectures are prepared as a contribution to the NATO Advanced Study In- stitute held in Ohrid, Macedonia, in August 2014. The topic of the conference was on the arithmetic of superelliptic curves, and this lecture will focus on the Weierstrass points of such curves. Since the Weierstrass points are an important concept of the theory of Riemann surfaces and algebraic curves and serve as a prerequisite for studying the automorphisms of the curves we will give in this lecture a detailed account of holomorphic and meromorphic functions of Riemann arXiv:1502.06285v1 [math.CV] 22 Feb 2015 surfaces, the proofs of the Weierstrass and Noether gap theorems, the Riemann- Hurwitz theorem, and the basic definitions and properties of higher-order Weier- strass points. Weierstrass points of algebraic curves are defined as a consequence of the important theorem of Riemann-Roch in the theory of algebraic curves. As an immediate application, the set of Weierstrass points is an invariant of a curve which is useful in the study of the curve’s automorphism group and the fixed points of automorphisms. In Part 1 we cover some of the basic material on Riemann surfaces and algebraic curves and their Weierstrass points. -
Weierstrass Points on Cyclic Covers of the Projective Line 3357
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 8, August 1996 WEIERSTRASS POINTS ON CYCLIC COVERS OFTHEPROJECTIVELINE CHRISTOPHER TOWSE Abstract. We are interested in cyclic covers of the projective line which are totally ramified at all of their branch points. We begin with curves given by an equation of the form yn = f(x), where f is a polynomial of degree d. Under a mild hypothesis, it is easy to see that all of the branch points must be Weierstrass points. Our main problem is to find the total Weierstrass weight of these points, BW. We obtain a lower bound for BW,whichweshowisexact if n and d are relatively prime. As a fraction of the total Weierstrass weight of all points on the curve, we get the following particularly nice asymptotic formula (as well as an interesting exact formula): BW n +1 lim = , d g3 g 3(n 1)2 →∞ − − where g is the genus of the curve. In the case that n = 3 (cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes p, the branch points and the non-branch Weierstrass points remain distinct modulo p. Introduction A Weierstrass point is a point on a curve such that there exists a non-constant function which has a low order pole at the point and no other poles. By low order we mean that the pole has order less than or equal to the genus g of the curve. The Riemann-Roch theorem shows that every point on a curve has a (non-constant) function associated to it which has a pole of order less than or equal to g +1and no other poles, so Weierstrass points are somewhat special. -
Holomorphic Embedding of Complex Curves in Spaces of Constant Holomorphic Curvature (Wirtinger's Theorem/Kaehler Manifold/Riemann Surfaces) ISSAC CHAVEL and HARRY E
Proc. Nat. Acad. Sci. USA Vol. 69, No. 3, pp. 633-635, March 1972 Holomorphic Embedding of Complex Curves in Spaces of Constant Holomorphic Curvature (Wirtinger's theorem/Kaehler manifold/Riemann surfaces) ISSAC CHAVEL AND HARRY E. RAUCH* The City College of The City University of New York and * The Graduate Center of The City University of New York, 33 West 42nd St., New York, N.Y. 10036 Communicated by D. C. Spencer, December 21, 1971 ABSTRACT A special case of Wirtinger's theorem ever, the converse is not true in general: the complex curve asserts that a complex curve (two-dimensional) hob-o embedded as a real minimal surface is not necessarily holo- morphically embedded in a Kaehler manifold is a minimal are obstructions. With the by morphically embedded-there surface. The converse is not necessarily true. Guided that we view the considerations from the theory of moduli of Riemann moduli problem in mind, this fact suggests surfaces, we discover (among other results) sufficient embedding of our complex curve as a real minimal surface topological aind differential-geometric conditions for a in a Kaehler manifold as the solution to the differential-geo- minimal (Riemannian) immersion of a 2-manifold in for our present mapping problem metric metric extremal problem complex projective space with the Fubini-Study as our infinitesimal moduli, differential- to be holomorphic. and that we then seek, geometric invariants on the curve whose vanishing forms neces- sary and sufficient conditions for the minimal embedding to be INTRODUCTION AND MOTIVATION holomorphic. Going further, we observe that minimal surface It is known [1-5] how Riemann's conception of moduli for con- evokes the notion of second fundamental forms; while the formal mapping of homeomorphic, multiply connected Rie- moduli problem, again, suggests quadratic differentials. -
Abelian Integrals and the Genesis of Abel's Greatest Discovery
Abelian integrals and the genesis of Abel's greatest discovery Christian F. Skau Department of Mathematical Sciences, Norwegian University of Science and Technology March 20, 2020 Draft P(x) R(x) = Q(x) 3x 7 − 18x 6 + 46x 5 − 84x 4 + 127x 3 − 122x 2 + 84x − 56 = (x − 2)3(x + i)(x − i) 4x 4 − 12x 3 + 7x 2 − x + 1 = 3x 2 + 7 + (x − 2)3(x + i)(x − i) (−1) 3 4 i=2 (−i=2) = 3x 2 + 7 + + + + + (x − 2)3 (x − 2)2 x − 2 x + i x − i Draft Z 1=2 −3 R(x) dx = x 3 + 7x + + + 4 log (x − 2) (x − 2)2 x − 2 i −i + log (x + i) + log (x − i) 2 2 i −i = Re(x) + 4 log (x − 2) + log (x + i) + log (x − i) 2 2 The integral of a rational function R(x) is a rational function Re(x) plus a sum of logarithms of the form log(x + a), a 2 C, multiplied with constants. Draft eix − e−ix 1 p sin x = arcsin x = log ix + 1 − x 2 2i i Trigonometric functions and the inverse arcus-functions can be expressed by (complex) exponential and logarithmic functions. Draft A rational function R(x; y) of x and y is of the form P(x; y) R(x; y) = ; Q(x; y) where P and Q are polynomials in x and y. So, P i j aij x y R(x; y) = P k l bkl x y Similarly we define a rational function of x1; x2;:::; xn. -
Curvature Theorems on Polar Curves
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES Volume 33 March 15, 1947 Number 3 CURVATURE THEOREMS ON POLAR CUR VES* By EDWARD KASNER AND JOHN DE CICCO DEPARTMENT OF MATHEMATICS, COLUMBIA UNIVERSITY, NEW YoRK Communicated February 4, 1947 1. The principle of duality in projective geometry is based on the theory of poles and polars with respect to a conic. For a conic, a point has only one kind of polar, the first polar, or polar straight line. However, for a general algebraic curve of higher degree n, a point has not only the first polar of degree n - 1, but also the second polar of degree n - 2, the rth polar of degree n - r, :. ., the (n - 1) polar of degree 1. This last polar is a straight line. The general polar theory' goes back to Newton, and was developed by Bobillier, Cayley, Salmon, Clebsch, Aronhold, Clifford and Mayer. For a given curve Cn of degree n, the first polar of any point 0 of the plane is a curve C.-1 of degree n - 1. If 0 is a point of C, it is well known that the polar curve C,,- passes through 0 and touches the given curve Cn at 0. However, the two curves do not have the same curvature. We find that the ratio pi of the curvature of C,- i to that of Cn is Pi = (n -. 2)/(n - 1). For example, if the given curve is a cubic curve C3, the first polar is a conic C2, and at an ordinary point 0 of C3, the ratio of the curvatures is 1/2.