Proceedings of XIII Fall Workshop on Geometry and Physics

Total Page:16

File Type:pdf, Size:1020Kb

Proceedings of XIII Fall Workshop on Geometry and Physics PUBLICACIONES de la Real Sociedad Matem´atica Espa˜nola Volumen 9 Proceedings of XIII Fall Workshop on Geometry and Physics Murcia, Spain September 20–22, 2004 Edited by: Luis J. Al´ıas Linares Angel Ferr´andez Izquierdo Mar´ıa Angeles Hern´andez Cifre Pascual Lucas Saor´ın Jos´e Antonio Pastor Gonz´alez Organizing Committee Luis J. Al´ıas Linares ([email protected]) Pablo Miguel Chac´on Mart´ın ([email protected]) Angel Ferr´andez Izquierdo ([email protected]) Mar´ıa Angeles Hern´andez Cifre ([email protected]) Pascual Lucas Saor´ın ([email protected]) Pablo Mira Carrillo ([email protected]) Jos´e Antonio Pastor Gonz´alez ([email protected]) All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, whithout written permission of the Editors. L.J. Al´ıas, A. Ferr´andez, M.A. Hern´andez, P. Lucas, J.A. Pastor, editors Proceedings of XIII Fall Workshop on Geometry and Physics ISBN: 84-933610-6-2 Dep´osito legal: MU-1816-2005 Impreso en Espa˜na - Printed in Spain List of contributions Mini-Course J. Fernando Barbero G. Introductiontogeneralrelativity...................... 3 Invited lectures AnaBelaCruzeiro Some probabilistic estimates of heat kernel derivatives and applications 45 Eric Loubeau An aper¸cuofharmonicmapsandmorphisms............... 55 Juan Carlos Marrero and Diana Sosa Tulczyjew’s triple associated with a Lie affgebroid and a Hamiltonian section.................................. 66 Elena Medina and Mar´ıa Jos´eMar´ın New solutions of the Nizhnik-Novikov-Veselov model through symmetry transformations............................. 80 Stefano Montaldo and Irene I. Onnis Invariant surfaces in H2 × R with constant (mean or Gauss) curvature . 91 Jeanette E. Nelson and Roger F. Picken Classical and quantum geometry of moduli spaces in three-dimensional gravity.................................. 104 Estelita G.L.R. Vaz Onwarpedanddoublewarpedspace-times................ 115 Jean-Claude Zambrini Brownianmotionandsymplecticgeometry................ 126 Communications Antonio Ca˜nete Least-perimeterpartitionsofthedisk................... 141 Isabel Fern´andez and Francisco J. L´opez Periodic maximal graphs in the Lorentz-Minkowki space L3 ....... 149 4 Benito Hern´andez–Bermejo Hamiltonization and the study of Jacobi equations: some perspectives andresults................................ 155 Josep Llosa and Daniel Soler SemiRiemannian metrics as deformations of a constant curvature metric 161 Antonio L´opez Almorox Cohomology of Lie groups associated to compatible linear cocycles and Lie group structures on T ∗G ...................... 167 Jos´e M.M. Senovilla TrappedsubmanifoldsinLorentziangeometry.............. 175 Posters Rui Albuquerque Riemannian geometry of the twistor space of a symplectic manifold . 183 Angel Ballesteros, Francisco J. Herranz and Orlando Ragnisco q-Poisson coalgebras and integrable dynamics on spaces with variable curvature................................. 189 J. Fernando Barbero, Guillermo A. Mena Marug´an and Eduardo J.S. Villase˜nor Perturbative and non-perturbative quantum Einstein-Rosen waves . 195 Manuel Calixto Generalized W∞ higher-spinalgebrasandquantization.......... 202 Fernando Etayo, Rafael Santamar´ıa and Uju´eR.Tr´ıas Triplestructuresonamanifold....................... 208 Xavier Gr`acia and Rub´en Mart´ın Vector hulls and jet bundles . .................... 215 Julio Guerrero and Jos´eManuelP´erez Quantization of the regularized Kepler and related problems . 221 David Iglesias, Juan Carlos Marrero, Edith Padr´on and Diana Sosa Compatibility between Lie affgebroid and affine Jacobi structures on an affine bundle . .............................. 227 Manuel de Le´on, Jes´us Mar´ın-Solano, Juan Carlos Marrero, Miguel C. Mu˜noz-Lecanda and Narciso Rom´an-Roy Constraintalgorithmforsingularfieldtheories.............. 233 Eva Miranda A normal form theorem for integrable systems on contact manifolds . 240 C´esar Rosales Isoperimetric problems inside Euclidean convex domains . 247 Preface The professional dialogue between Mathematicians and Physicists is some- thing as natural as enriching in the daily life of a secondary school teacher. It seems that it is in the following educational step, the university, where the distance begins to be significant, situation that becomes usual as the specializ- ation level grows. Identical situation occurs in the realm of own Physicists or Mathematicians of the different areas, a classical characteristic of the Spanish people that we already see almost natural that a topologist and a statistician nothing must say themselves mathematically speaking. Perhaps it is exagger- ated, but it is what we daily observe. Luckily, who we define ourselves as geometricians or physicists had been thirteen years already trying to be the exception to so displeased rule and in that time giant steps have occurred. We have no doubt that we continue undergoing wide gaps of understanding, but every year a shortening of that ancestral distance is clearly observed. We think that the convergence is going to be achieved, essentially, through the Differential Geometry tools, mainly with theoretical physicists. It turns also out to be true that any other area of Mathematics can contribute in the same way to the progress, not only of Physics, but of any other science. The participants in these series of workshops have bet to foster a better understanding between Mathematicians and Physicists in order to undertake joint projects of greater scope, such as the society and the current research are demanding from them. We think that this is the right way to promote the Spanish science and these Proceedings will contribute to that aim. Further- more, they will be the best test of our commitment and support to Spanish quality publications. In Oviedo, where Murcia was nominated to organize this Workshop, and later in Coimbra, they left the strip in the highest, but we accepted the chal- 5 6 lenge to surpass it and we spared no effort to get it. We have done one’s bit to shorten the distance between geometry and physics; and what is better, to increase the quality, all over the world recognized, of those working in these areas. We take one’s chance to reiterate our public gratefulness, by its commend- able perseverance and example, to those who were very deserving of a ham, the colleagues Eduardo Mart´ınez Fern´andez, Miguel Carlos Mu˜noz Lecanda and Narciso Rom´an Roy. The University of Murcia, the S´eneca Foundation, the Spanish Ministry of Education and Science, the Cajamurcia Foundation, the Caja de Ahorros del Mediterr´aneo and the Real Spanish Mathematical Society made the Workshop and these Proceedings possible, so that they deserve our more sincere grate- fulness. As well as to all those that participated either learning, or teaching or, simply, engaging in a dialogue. Murcia, September 2005 The Editors Luis J. Al´ıas Linares Angel Ferr´andez Izquierdo Mar´ıa Angeles Hern´andez Cifre Pascual Lucas Saor´ın Jos´e Antonio Pastor Gonz´alez List of participants Name (Institution) e-mail address Alma-Luisa ALBUJER BROTONS (Univ. Murcia) [email protected] Rui Pedro ALBUQUERQUE (Univ. Evora)´ [email protected] Luis Jos´eAL´IAS LINARES (Univ. Murcia) [email protected] Luis Carlos de ANDRES´ DOMINGO (Univ. Pa´ıs Vasco) [email protected] Josu ARROYO OLEA (Univ. Pa´ıs Vasco) [email protected] Mar´ıa BARBERO LIN˜AN´ (Univ. Polit`ecnica de Catalunya) Manuel BARROS D´IAZ (Univ. Granada) [email protected] Roberto BENITO PRIETO (CSIC) rbenito@imaff.cfmac.csic.es Michele BENYOUNES (CNRS, Univ. Brest) [email protected] Miguel BROZOS VAZQUEZ´ (Univ. Santiago de Compostela) [email protected] Mar´ıa Teresa de BUSTOS MUNOZ˜ (Univ. Salamanca) Jos´e Luis CABRERIZO JARA´IZ (Univ. Sevilla) [email protected] Manuel CALIXTO MOLINA (Univ. Polit´ecnica de Cartagena) [email protected] Antonio Jes´us CANETE˜ MART´IN (Univ. Granada) [email protected] Jos´e Fernando CARINENA˜ MARZO (Univ. Zaragoza) [email protected] Alexandre CORTES´ AYASO (Univ. Santiago de Compostela) Ana Bela CRUZEIRO (Inst. Sup. T´ecnico, Lisboa) [email protected] Marina DELGADO TELLEZ´ DE CEPEDA (Univ. Carlos III) [email protected] Pablo Miguel CHACON´ MART´IN (Univ. Salamanca) [email protected] Alberto ENCISO CARRASCO (Univ. Complutense de Madrid) aenciso@fis.ucm.es Laura ESTEVE CORBALAN´ (Univ. Valencia) Fernando ETAYO GORDEJUELA (Univ. Cantabria) [email protected] Ra´ul Manuel FALCON´ GANFORNINA (Univ. Sevilla) [email protected] Rui Loja FERNANDES (Inst. Sup. T´ecnico, Lisboa) [email protected] Manuel FERNANDEZ´ ANDRES´ (Univ. Sevilla) [email protected] Isabel FERNANDEZ´ DELGADO (Univ. Granada) [email protected] Antonio FERNANDEZ´ MART´INEZ (Univ. Salamanca) [email protected] Jos´eFERNANDEZ´ NU´NEZ˜ (Univ. Oviedo) [email protected] Angel FERRANDEZ´ IZQUIERDO (Univ. Murcia) [email protected] Jos´e Antonio FERRY RUIZ [email protected] Carlos Alberto FUERTES PASCUAL (Univ. Salamanca) Pedro Luis GARC´IA PEREZ´ (Univ. Salamanca) [email protected] Xavier GRACIA` SABATE´ (Univ. Polit`ecnica de Catalunya) [email protected] Julio GUERRERO GARC´IA (Univ. Murcia) [email protected] Benito HERNANDEZ´ BERMEJO (Univ. Rey Juan Carlos) [email protected] Mar´ıa Angeles´ HERNANDEZ´ CIFRE (Univ. Murcia) [email protected] Pedro Jose HERRERO PINEYRO˜ (Univ. Murcia) [email protected] 8 Ana Mar´ıa HURTADO CORTEGANA (Univ. Valencia) [email protected] Miguel Angel´ JAVALOYES VICTORIA (Univ. Sao Paulo) [email protected]
Recommended publications
  • MAT 362 at Stony Brook, Spring 2011
    MAT 362 Differential Geometry, Spring 2011 Instructors' contact information Course information Take-home exam Take-home final exam, due Thursday, May 19, at 2:15 PM. Please read the directions carefully. Handouts Overview of final projects pdf Notes on differentials of C1 maps pdf tex Notes on dual spaces and the spectral theorem pdf tex Notes on solutions to initial value problems pdf tex Topics and homework assignments Assigned homework problems may change up until a week before their due date. Assignments are taken from texts by Banchoff and Lovett (B&L) and Shifrin (S), unless otherwise noted. Topics and assignments through spring break (April 24) Solutions to first exam Solutions to second exam Solutions to third exam April 26-28: Parallel transport, geodesics. Read B&L 8.1-8.2; S2.4. Homework due Tuesday May 3: B&L 8.1.4, 8.2.10 S2.4: 1, 2, 4, 6, 11, 15, 20 Bonus: Figure out what map projection is used in the graphic here. (A Facebook account is not needed.) May 3-5: Local and global Gauss-Bonnet theorem. Read B&L 8.4; S3.1. Homework due Tuesday May 10: B&L: 8.1.8, 8.4.5, 8.4.6 S3.1: 2, 4, 5, 8, 9 Project assignment: Submit final version of paper electronically to me BY FRIDAY MAY 13. May 10: Hyperbolic geometry. Read B&L 8.5; S3.2. No homework this week. Third exam: May 12 (in class) Take-home exam: due May 19 (at presentation of final projects) Instructors for MAT 362 Differential Geometry, Spring 2011 Joshua Bowman (main instructor) Office: Math Tower 3-114 Office hours: Monday 4:00-5:00, Friday 9:30-10:30 Email: joshua dot bowman at gmail dot com Lloyd Smith (grader Feb.
    [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]
  • Geometry and Entropies in a Fixed Conformal Class on Surfaces
    GEOMETRY AND ENTROPIES IN A FIXED CONFORMAL CLASS ON SURFACES THOMAS BARTHELME´ AND ALENA ERCHENKO Abstract. We show the flexibility of the metric entropy and obtain additional restrictions on the topological entropy of geodesic flow on closed surfaces of negative Euler characteristic with smooth non-positively curved Riemannian metrics with fixed total area in a fixed conformal class. Moreover, we obtain a collar lemma, a thick-thin decomposition, and precompactness for the considered class of metrics. Also, we extend some of the results to metrics of fixed total area in a fixed conformal class with no focal points and with some integral bounds on the positive part of the Gaussian curvature. 1. Introduction When M is a fixed surface, there has been a long history of studying how the geometric or dynamical data (e.g., the Laplace spectrum, systole, entropies or Lyapunov exponents of the geodesic flow) varies when one varies the metric on M, possibly inside a particular class. In [BE20], we studied these questions in a class of metrics that seemed to have been overlooked: the family of non-positively curved metrics within a fixed conformal class. In this article, we prove several conjectures made in [BE20], as well as give a fairly complete, albeit coarse, picture of the geometry of non-positively curved metrics within a fixed conformal class. Since Gromov’s famous systolic inequality [Gro83], there has been a lot of interest in upper bounds on the systole (see for instance [Gut10]). In general, there is no positive lower bound on the systole. However, we prove here that non-positively curved metrics in a fixed conformal class do admit such a lower bound.
    [Show full text]
  • On the Quantization Problem in Curved Space
    Wright State University CORE Scholar Browse all Theses and Dissertations Theses and Dissertations 2012 On the Quantization Problem in Curved Space Benjamin Bernard Wright State University Follow this and additional works at: https://corescholar.libraries.wright.edu/etd_all Part of the Physics Commons Repository Citation Bernard, Benjamin, "On the Quantization Problem in Curved Space" (2012). Browse all Theses and Dissertations. 1094. https://corescholar.libraries.wright.edu/etd_all/1094 This Thesis is brought to you for free and open access by the Theses and Dissertations at CORE Scholar. It has been accepted for inclusion in Browse all Theses and Dissertations by an authorized administrator of CORE Scholar. For more information, please contact [email protected]. ON THE QUANTIZATION PROBLEM IN CURVED SPACE A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science by BENJAMIN JOSEPH BERNARD B.S., University of Cincinnati, 1999 2012 Wright State University WRIGHT STATE UNIVERSITY GRADUATE SCHOOL July 3, 2012 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPERVISION BY Benjamin Joseph Bernard ENTITLED On the Quantization Problem in Curved Space BE ACCEPTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science. Lok C. Lew Yan Voon, Ph.D., Thesis Director Doug Petkie, Ph.D. Chair, Department of Physics Committee on Final Examination Lok C. Lew Yan Voon, Ph.D. Morten Willatzen, Ph.D. Gary Farlow, Ph.D. Andrew Hsu, Ph.D. Dean, Graduate School ABSTRACT Bernard, Benjamin Joseph. M.S., Department of Physics, Wright State Univer- sity, 2012. On the Quantization Problem in Curved Space The nonrelativistic quantum mechanics of particles constrained to curved surfaces is studied.
    [Show full text]
  • Arxiv:1707.09532V1 [Math.DG] 29 Jul 2017
    TRACTORS AND TRACTRICES IN RIEMANNIAN MANIFOLDS JESPER J. MADSEN AND STEEN MARKVORSEN Abstract. We generalize the notion of planar bicycle tracks { a.k.a. one-trailer systems { to so-called tractor/tractrix systems in general Riemannian manifolds and prove explicit expressions for the length of the ensuing tractrices and for the area of the do- mains that are swept out by any given tractor/tractrix system. These expressions are sensitive to the curvatures of the ambient Riemannian manifold, and we prove explicit estimates for them based on Rauch's and Toponogov's comparison theorems. More- over, the general length shortening property of tractor/tractrix sys- tems is used to generate geodesics in homotopy classes of curves in the ambient manifold. 1. Introduction The classical tractrix curve appears in virtually every textbook on differential geometry of curves and surfaces { see e.g. [6], [16], [31, p. 239], [21, p. 67], and figure 9 below. The tractrix has a long and fasci- nating history beginning with works of Huygens, Leibniz, Newton, and Euler, see [3]. Not to mention the celebrated watch track experiment by Claude Perrault, long before the invention of the bicycle { see [12], [7], [11]. We quote from [3, p. 1065]:\At a meeting in Paris in 1693 Claude Perrault laid his watch on the table, with the long chain drawn out in a straight line ([4, vol. 3]). He showed that when he moved the end of the chain along a straight line, keeping the chain taut, the watch was dragged along a certain curve. This was one of the early demonstra- tions of the tractrix." arXiv:1707.09532v1 [math.DG] 29 Jul 2017 Moreover, this classical tractrix gives rise to interesting classical sur- faces as well: When rotated around the axis along which the tractrix is pulled, any regular segment of the tractrix curve generates a pseudo- sphere of constant negative Gauss curvature, and the involute of the full classical tractrix curve, including its cusp singularity, is a catenary, which itself, when rotated around the axis, generates a catenoid, i.e.
    [Show full text]
  • Mathematics of the Gateway Arch Page 220
    ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Highlights in Springer’s eBook of the American Mathematical Society Collection February 2010 Volume 57, Number 2 An Invitation to Cauchy-Riemann NEW 4TH NEW NEW EDITION and Sub-Riemannian Geometries 2010. XIX, 294 p. 25 illus. 4th ed. 2010. VIII, 274 p. 250 2010. XII, 475 p. 79 illus., 76 in 2010. XII, 376 p. 8 illus. (Copernicus) Dustjacket illus., 6 in color. Hardcover color. (Undergraduate Texts in (Problem Books in Mathematics) page 208 ISBN 978-1-84882-538-3 ISBN 978-3-642-00855-9 Mathematics) Hardcover Hardcover $27.50 $49.95 ISBN 978-1-4419-1620-4 ISBN 978-0-387-87861-4 $69.95 $69.95 Mathematics of the Gateway Arch page 220 Model Theory and Complex Geometry 2ND page 230 JOURNAL JOURNAL EDITION NEW 2nd ed. 1993. Corr. 3rd printing 2010. XVIII, 326 p. 49 illus. ISSN 1139-1138 (print version) ISSN 0019-5588 (print version) St. Paul Meeting 2010. XVI, 528 p. (Springer Series (Universitext) Softcover ISSN 1988-2807 (electronic Journal No. 13226 in Computational Mathematics, ISBN 978-0-387-09638-4 version) page 315 Volume 8) Softcover $59.95 Journal No. 13163 ISBN 978-3-642-05163-0 Volume 57, Number 2, Pages 201–328, February 2010 $79.95 Albuquerque Meeting page 318 For access check with your librarian Easy Ways to Order for the Americas Write: Springer Order Department, PO Box 2485, Secaucus, NJ 07096-2485, USA Call: (toll free) 1-800-SPRINGER Fax: 1-201-348-4505 Email: [email protected] or for outside the Americas Write: Springer Customer Service Center GmbH, Haberstrasse 7, 69126 Heidelberg, Germany Call: +49 (0) 6221-345-4301 Fax : +49 (0) 6221-345-4229 Email: [email protected] Prices are subject to change without notice.
    [Show full text]
  • DIFFERENTIAL GEOMETRY and the UPPER HALF PLANE and DISK MODELS of the LOBACHEVSKI Or HYPERBOLIC PLANE Book 1 William Schulz1 1
    DIFFERENTIAL GEOMETRY AND THE UPPER HALF PLANE AND DISK MODELS OF THE LOBACHEVSKI or HYPERBOLIC PLANE Book 1 William Schulz1 Department of Mathematics and Statistics Northern Arizona University, Flagstaff, AZ 86011 1. INTRODUCTION This module has a number of goals which are not entirely consistent with one another. Our first objective is to develop the initial stages of Lobachevski ge- ometry in as natural a manner as possible given tools which are commonly available. These are elementary complex analysis and elementary differential geometry. The second goal is to use Lobachevski geometry as an example of a Riemannian Geometry. In some sense it is a maximally simple example since it is not a surface embeddable in 3 space but it is still two dimensional and topologically trivial. Hence it can serve as a sandbox for learning concepts such as geodesic curvature and parallel translation in a context where these concepts are not totally trivial but still not very difficult. Third, by considering the upper half plane model and the disk model we have a good example of how easy things can be when placed in an appropriate setting. We also have a non-trivial and splendidly useful example of an isometry. The novice at Lobachevski geometry would profit from looking at the in- toroductory material which shows pictures of the upper half plane model (Book 0). A word of caution. Much of the material I develop here can be developed in a more elementary manner, and some persons, for example philosophy stu- dents, might be more comfortable going that route, in which case there are many good books available.
    [Show full text]
  • The Second Fundamental Form. Geodesics. the Curvature Tensor
    Differential Geometry Lia Vas The Second Fundamental Form. Geodesics. The Curvature Tensor. The Fundamental Theorem of Surfaces. Manifolds The Second Fundamental Form and the Christoffel symbols. Consider a surface x = @x @x x(u; v): Following the reasoning that x1 and x2 denote the derivatives @u and @v respectively, we denote the second derivatives @2x @2x @2x @2x @u2 by x11; @v@u by x12; @u@v by x21; and @v2 by x22: The terms xij; i; j = 1; 2 can be represented as a linear combination of tangential and normal component. Each of the vectors xij can be repre- sented as a combination of the tangent component (which itself is a combination of vectors x1 and x2) and the normal component (which is a multi- 1 2 ple of the unit normal vector n). Let Γij and Γij denote the coefficients of the tangent component and Lij denote the coefficient with n of vector xij: Thus, 1 2 P k xij = Γijx1 + Γijx2 + Lijn = k Γijxk + Lijn: The formula above is called the Gauss formula. k The coefficients Γij where i; j; k = 1; 2 are called Christoffel symbols and the coefficients Lij; i; j = 1; 2 are called the coefficients of the second fundamental form. Einstein notation and tensors. The term \Einstein notation" refers to the certain summation convention that appears often in differential geometry and its many applications. Consider a formula can be written in terms of a sum over an index that appears in subscript of one and superscript of P k the other variable. For example, xij = k Γijxk + Lijn: In cases like this the summation symbol is omitted.
    [Show full text]
  • Differential Geometry
    18.950 (Differential Geometry) Lecture Notes Evan Chen Fall 2015 This is MIT's undergraduate 18.950, instructed by Xin Zhou. The formal name for this class is “Differential Geometry". The permanent URL for this document is http://web.evanchen.cc/ coursework.html, along with all my other course notes. Contents 1 September 10, 20153 1.1 Curves...................................... 3 1.2 Surfaces..................................... 3 1.3 Parametrized Curves.............................. 4 2 September 15, 20155 2.1 Regular curves................................. 5 2.2 Arc Length Parametrization.......................... 6 2.3 Change of Orientation............................. 7 3 2.4 Vector Product in R ............................. 7 2.5 OH GOD NO.................................. 7 2.6 Concrete examples............................... 8 3 September 17, 20159 3.1 Always Cauchy-Schwarz............................ 9 3.2 Curvature.................................... 9 3.3 Torsion ..................................... 9 3.4 Frenet formulas................................. 10 3.5 Convenient Formulas.............................. 11 4 September 22, 2015 12 4.1 Signed Curvature................................ 12 4.2 Evolute of a Curve............................... 12 4.3 Isoperimetric Inequality............................ 13 5 September 24, 2015 15 5.1 Regular Surfaces................................ 15 5.2 Special Cases of Manifolds .......................... 15 6 September 29, 2015 and October 1, 2015 17 6.1 Local Parametrizations ...........................
    [Show full text]
  • 612 CLASS LECTURE: HYPERBOLIC GEOMETRY 1. Conformal Metrics As a Vector Space, C Has a Canonical Norm, the Same As the Standard
    612 CLASS LECTURE: HYPERBOLIC GEOMETRY JOSHUA P. BOWMAN 1. Conformal metrics As a vector space, C has a canonical norm, the same as the standard R2 norm. Denote this jdzj|one should think of dz as the identity map on C: ζ 7! ζ. This also means C has a standard norm as the tangent space to itself at any of its points. We will consistently use Roman letters (e.g., z) to represent a point in C when we're thinking of it as a topological ◦ space, and Greek letters (e.g., ζ) to represent a point in C thought of as a vector. Let U ⊂ C be a plane domain (non-empty, connected, but not necessarily simply connected). Definition 1.1. A conformal metric on U is a metric of the form ρ jdzj, where ρ is a smooth, positive function on U. We will also use ρ to denote the metric itself, not just the function. Almost always, \conformal" is used with respect to something else; here, we mean that angles in a conformal metric are the same as those in the background Euclidean metric. In other words, the unit ball of the norm in the tangent space at z is invariant under (Euclidean) rotations. Example 1.2. On U = C n f0g, ρ = jdzj=jzj is a conformal metric. U is preserved by multiplication by a non-zero complex number a; call such a map ma. The derivative of ma is again multiplication by a. Let z 2 U and let ζ 2 Tz(U) = C.
    [Show full text]
  • Riemannian Geometry
    Cambridge University Press 0521450543 - The Geometry of Total Curvature on Complete Open Surfaces Katsuhiro Shiohama, Takashi Shioya and Minoru Tanaka Excerpt More information 1 Riemannian geometry We introduce the basic tools of Riemannian geometry that we shall need as- suming that readers already know basic facts about manifolds. Only smooth manifolds are discussed unless otherwise mentioned. The use of a local chart (local coordinates) will be convenient for readers at the beginning. Tensor cal- culus is used to introduce geodesics, parallelism, covariant derivative and the Riemannian curvature tensor. In Sections 1.1 to 1.3 the Einstein convention will be adopted without further mention. However, it is not convenient for later discussion, for example, the second variation formula or Jacobi fields. To avoid confusion we shall use vector fields and connection forms to discuss such matters as Jacobi fields and conjugate points. 1.1 The Riemannian metric Let M be an n-dimensional, connected and smooth manifold and (U, x) local coordinates around a point p ∈ M. A point q ∈ U is expressed as x(q) = 1 ,..., n ∈ ⊂ n (x (q) x (q)) x(U) R . The tangent space to M at p is denoted by = Tp M or Mp and TM : p∈M Tp M denotes the tangent bundle over M.We denote by π : TM → M the projection map. Let X (M) and X (U) be the spaces of all smooth vector fields over M and U respectively and C∞(M) the space of all smooth functions over M. A positive definite smooth symmetric bilinear form g : X (M) × X (M) → C∞(M) is by definition a Riemannian metric over M.
    [Show full text]
  • 7. the Gauss-Bonnet Theorem
    Math 501 - Differential Geometry Herman Gluck Thursday March 29, 2012 7. THE GAUSS-BONNET THEOREM The Gauss-Bonnet Theorem is one of the most beautiful and one of the deepest results in the differential geometry of surfaces. It concerns a surface S with boundary S in Euclidean 3-space, and expresses a relation between: • the integral S K d(area) of the Gaussian curvature over the surface, • the integral S g ds of the geodesic curvature of the boundary of the surface, and • the topology of the surface, as expressed by its Euler characteristic: (S) = # Vertices — # Edges + # Faces . 1 Do you recognize this surface? 2 We will approach this subject in the following way: First we'll build up some experience with examples in which we integrate Gaussian curvature over surfaces and integrate geodesic curvature over curves. Then we'll state and explain the Gauss-Bonnet Theorem and derive a number of consequences. Next we'll try to understand on intuitive grounds why the Gauss-Bonnet Theorem is true. Finally, we'll prove the Gauss-Bonnet Theorem. 3 Johann Carl Friedrich Gauss (1777 - 1855), age 26 4 Examples of the Gauss-Bonnet Theorem. Round spheres of radius r. Gaussian curvature K = 1/r2 Area = 4 r2 2 2 S K d(area) = K Area = (1/r ) 4 r = 4 = 2 (S) . 5 Convex surfaces. Let S be any convex surface in 3-space. 6 Let N: S S2 be the Gauss map, with differential 2 dNp: TpS TN(p)S = TpS . The Gaussian curvature K(p) = det dNp is the local explosion factor for areas under the Gauss map.
    [Show full text]