A Historical Overview of Connections in Geometry A
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Connections on Bundles Md
Dhaka Univ. J. Sci. 60(2): 191-195, 2012 (July) Connections on Bundles Md. Showkat Ali, Md. Mirazul Islam, Farzana Nasrin, Md. Abu Hanif Sarkar and Tanzia Zerin Khan Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh, Email: [email protected] Received on 25. 05. 2011.Accepted for Publication on 15. 12. 2011 Abstract This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle , is called an affine connection on an -dimensional smooth manifold . By the general discussion of affine connection on vector bundles that necessarily exists on which is compatible with tensors. I. Introduction = < , > (2) In order to differentiate sections of a vector bundle [5] or where <, > represents the pairing between and ∗. vector fields on a manifold we need to introduce a Then is a section of , called the absolute differential structure called the connection on a vector bundle. For quotient or the covariant derivative of the section along . example, an affine connection is a structure attached to a differentiable manifold so that we can differentiate its Theorem 1. A connection always exists on a vector bundle. tensor fields. We first introduce the general theorem of Proof. Choose a coordinate covering { }∈ of . Since connections on vector bundles. Then we study the tangent vector bundles are trivial locally, we may assume that there is bundle. is a -dimensional vector bundle determine local frame field for any . By the local structure of intrinsically by the differentiable structure [8] of an - connections, we need only construct a × matrix on dimensional smooth manifold . each such that the matrices satisfy II. -
13 Circles and Cross Ratio
13 Circles and Cross Ratio Following up Klein’s Erlangen Program, after defining the group of linear transformations, we study the invariant properites of subsets in the Riemann Sphere under this group. Appolonius’ Circle Recall that a line or a circle on the complex plane, their corresondence in the Riemann Sphere are all circles. We would like to prove that any linear transformation f maps a circle or a line into another circle or line. To prove this, we may separate 4 cases: f maps a line to a line; f maps a line to a circle; f maps a circle to a line; and f maps a circle to a circle. Also, we observe that f is a composition of several simple linear transformations of three type: 1 f1(z)= az, f2(z)= a + b and f3(z)= z because az + b g(z) f(z)= = = g(z)φ(h(z)) (56) cz + d h(z) 1 where g(z) = az + b, h(z) = cz + d and φ(z) = z . Then it suffices to separate 8 cases: fj maps a line to a line; fj maps a line to a circle; fj maps a circle to a line; and fj maps a circle to a circle for j =1, 2, 3. To simplify this proof, we want to write a line or a circle by a single equation, although usually equations for a line and for a circle are are quite different. Let us consider the following single equation: z − z1 = k, (57) z − z2 for some z1, z2 ∈ C and 0 <k< ∞. -
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. -
2. Chern Connections and Chern Curvatures1
1 2. Chern connections and Chern curvatures1 Let V be a complex vector space with dimC V = n. A hermitian metric h on V is h : V £ V ¡¡! C such that h(av; bu) = abh(v; u) h(a1v1 + a2v2; u) = a1h(v1; u) + a2h(v2; u) h(v; u) = h(u; v) h(u; u) > 0; u 6= 0 where v; v1; v2; u 2 V and a; b; a1; a2 2 C. If we ¯x a basis feig of V , and set hij = h(ei; ej) then ¤ ¤ ¤ ¤ h = hijei ej 2 V V ¤ ¤ ¤ ¤ where ei 2 V is the dual of ei and ei 2 V is the conjugate dual of ei, i.e. X ¤ ei ( ajej) = ai It is obvious that (hij) is a hermitian positive matrix. De¯nition 0.1. A complex vector bundle E is said to be hermitian if there is a positive de¯nite hermitian tensor h on E. r Let ' : EjU ¡¡! U £ C be a trivilization and e = (e1; ¢ ¢ ¢ ; er) be the corresponding frame. The r hermitian metric h is represented by a positive hermitian matrix (hij) 2 ¡(; EndC ) such that hei(x); ej(x)i = hij(x); x 2 U Then hermitian metric on the chart (U; ') could be written as X ¤ ¤ h = hijei ej For example, there are two charts (U; ') and (V; Ã). We set g = à ± '¡1 :(U \ V ) £ Cr ¡¡! (U \ V ) £ Cr and g is represented by matrix (gij). On U \ V , we have X X X ¡1 ¡1 ¡1 ¡1 ¡1 ei(x) = ' (x; "i) = à ± à ± ' (x; "i) = à (x; gij"j) = gijà (x; "j) = gije~j(x) j j For the metric X ~ hij = hei(x); ej(x)i = hgike~k(x); gjle~l(x)i = gikhklgjl k;l that is h = g ¢ h~ ¢ g¤ 12008.04.30 If there are some errors, please contact to: [email protected] 2 Example 0.2 (Fubini-Study metric on holomorphic tangent bundle T 1;0Pn). -
Projective Coordinates and Compactification in Elliptic, Parabolic and Hyperbolic 2-D Geometry
PROJECTIVE COORDINATES AND COMPACTIFICATION IN ELLIPTIC, PARABOLIC AND HYPERBOLIC 2-D GEOMETRY Debapriya Biswas Department of Mathematics, Indian Institute of Technology-Kharagpur, Kharagpur-721302, India. e-mail: d [email protected] Abstract. A result that the upper half plane is not preserved in the hyperbolic case, has implications in physics, geometry and analysis. We discuss in details the introduction of projective coordinates for the EPH cases. We also introduce appropriate compactification for all the three EPH cases, which results in a sphere in the elliptic case, a cylinder in the parabolic case and a crosscap in the hyperbolic case. Key words. EPH Cases, Projective Coordinates, Compactification, M¨obius transformation, Clifford algebra, Lie group. 2010(AMS) Mathematics Subject Classification: Primary 30G35, Secondary 22E46. 1. Introduction Geometry is an essential branch of Mathematics. It deals with the proper- ties of figures in a plane or in space [7]. Perhaps, the most influencial book of all time, is ‘Euclids Elements’, written in around 3000 BC [14]. Since Euclid, geometry usually meant the geometry of Euclidean space of two-dimensions (2-D, Plane geometry) and three-dimensions (3-D, Solid geometry). A close scrutiny of the basis for the traditional Euclidean geometry, had revealed the independence of the parallel axiom from the others and consequently, non- Euclidean geometry was born and in projective geometry, new “points” (i.e., points at infinity and points with complex number coordinates) were intro- duced [1, 3, 9, 26]. In the eighteenth century, under the influence of Steiner, Von Staudt, Chasles and others, projective geometry became one of the chief subjects of mathematical research [7]. -
Proper Affine Actions and Geodesic Flows of Hyperbolic Surfaces
ANNALS OF MATHEMATICS Proper affine actions and geodesic flows of hyperbolic surfaces By William M. Goldman, Franc¸ois Labourie, and Gregory Margulis SECOND SERIES, VOL. 170, NO. 3 November, 2009 anmaah Annals of Mathematics, 170 (2009), 1051–1083 Proper affine actions and geodesic flows of hyperbolic surfaces By WILLIAM M. GOLDMAN, FRANÇOIS LABOURIE, and GREGORY MARGULIS Abstract 2 Let 0 O.2; 1/ be a Schottky group, and let † H =0 be the corresponding D hyperbolic surface. Let Ꮿ.†/ denote the space of unit length geodesic currents 1 on †. The cohomology group H .0; V/ parametrizes equivalence classes of affine deformations u of 0 acting on an irreducible representation V of O.2; 1/. We 1 define a continuous biaffine map ‰ Ꮿ.†/ H .0; V/ R which is linear on 1 W ! the vector space H .0; V/. An affine deformation u acts properly if and only if ‰.; Œu/ 0 for all Ꮿ.†/. Consequently the set of proper affine actions ¤ 2 whose linear part is a Schottky group identifies with a bundle of open convex cones 1 in H .0; V/ over the Fricke-Teichmüller space of †. Introduction 1. Hyperbolic geometry 2. Affine geometry 3. Flat bundles associated to affine deformations 4. Sections and subbundles 5. Proper -actions and proper R-actions 6. Labourie’s diffusion of Margulis’s invariant 7. Nonproper deformations 8. Proper deformations References Goldman gratefully acknowledges partial support from National Science Foundation grants DMS- 0103889, DMS-0405605, DMS-070781, the Mathematical Sciences Research Institute and the Oswald Veblen Fund at the Insitute for Advanced Study, and a Semester Research Award from the General Research Board of the University of Maryland. -
A Geodesic Connection in Fréchet Geometry
A geodesic connection in Fr´echet Geometry Ali Suri Abstract. In this paper first we propose a formula to lift a connection on M to its higher order tangent bundles T rM, r 2 N. More precisely, for a given connection r on T rM, r 2 N [ f0g, we construct the connection rc on T r+1M. Setting rci = rci−1 c, we show that rc1 = lim rci exists − and it is a connection on the Fr´echet manifold T 1M = lim T iM and the − geodesics with respect to rc1 exist. In the next step, we will consider a Riemannian manifold (M; g) with its Levi-Civita connection r. Under suitable conditions this procedure i gives a sequence of Riemannian manifolds f(T M, gi)gi2N equipped with ci c1 a sequence of Riemannian connections fr gi2N. Then we show that r produces the curves which are the (local) length minimizer of T 1M. M.S.C. 2010: 58A05, 58B20. Key words: Complete lift; spray; geodesic; Fr´echet manifolds; Banach manifold; connection. 1 Introduction In the first section we remind the bijective correspondence between linear connections and homogeneous sprays. Then using the results of [6] for complete lift of sprays, we propose a formula to lift a connection on M to its higher order tangent bundles T rM, r 2 N. More precisely, for a given connection r on T rM, r 2 N [ f0g, we construct its associated spray S and then we lift it to a homogeneous spray Sc on T r+1M [6]. Then, using the bijective correspondence between connections and sprays, we derive the connection rc on T r+1M from Sc. -
Lie Group and Geometry on the Lie Group SL2(R)
INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear -
Elementary Differential Geometry
ELEMENTARY DIFFERENTIAL GEOMETRY YONG-GEUN OH { Based on the lecture note of Math 621-2020 in POSTECH { Contents Part 1. Riemannian Geometry 2 1. Parallelism and Ehresman connection 2 2. Affine connections on vector bundles 4 2.1. Local expression of covariant derivatives 6 2.2. Affine connection recovers Ehresmann connection 7 2.3. Curvature 9 2.4. Metrics and Euclidean connections 9 3. Riemannian metrics and Levi-Civita connection 10 3.1. Examples of Riemannian manifolds 12 3.2. Covariant derivative along the curve 13 4. Riemann curvature tensor 15 5. Raising and lowering indices and contractions 17 6. Geodesics and exponential maps 19 7. First variation of arc-length 22 8. Geodesic normal coordinates and geodesic balls 25 9. Hopf-Rinow Theorem 31 10. Classification of constant curvature surfaces 33 11. Second variation of energy 34 Part 2. Symplectic Geometry 39 12. Geometry of cotangent bundles 39 13. Poisson manifolds and Schouten-Nijenhuis bracket 42 13.1. Poisson tensor and Jacobi identity 43 13.2. Lie-Poisson space 44 14. Symplectic forms and the Jacobi identity 45 15. Proof of Darboux' Theorem 47 15.1. Symplectic linear algebra 47 15.2. Moser's deformation method 48 16. Hamiltonian vector fields and diffeomorhpisms 50 17. Autonomous Hamiltonians and conservation law 53 18. Completely integrable systems and action-angle variables 55 18.1. Construction of angle coordinates 56 18.2. Construction of action coordinates 57 18.3. Underlying geometry of the Hamilton-Jacobi method 61 19. Lie groups and Lie algebras 62 1 2 YONG-GEUN OH 20. Group actions and adjoint representations 67 21. -
Geometric Control of Mechanical Systems Modeling, Analysis, and Design for Simple Mechanical Control Systems
Francesco Bullo and Andrew D. Lewis Geometric Control of Mechanical Systems Modeling, Analysis, and Design for Simple Mechanical Control Systems – Supplementary Material – August 1, 2014 Contents S1 Tangent and cotangent bundle geometry ................. S1 S1.1 Some things Hamiltonian.......................... ..... S1 S1.1.1 Differential forms............................. .. S1 S1.1.2 Symplectic manifolds .......................... S5 S1.1.3 Hamiltonian vector fields ....................... S6 S1.2 Tangent and cotangent lifts of vector fields .......... ..... S7 S1.2.1 More about the tangent lift ...................... S7 S1.2.2 The cotangent lift of a vector field ................ S8 S1.2.3 Joint properties of the tangent and cotangent lift . S9 S1.2.4 The cotangent lift of the vertical lift ............ S11 S1.2.5 The canonical involution of TTQ ................. S12 S1.2.6 The canonical endomorphism of the tangent bundle . S13 S1.3 Ehresmann connections induced by an affine connection . S13 S1.3.1 Motivating remarks............................ S13 S1.3.2 More about vector and fiber bundles .............. S15 S1.3.3 Ehresmann connections ......................... S17 S1.3.4 Linear connections and linear vector fields on vector bundles ....................................... S18 S1.3.5 The Ehresmann connection on πTM : TM M associated with a second-order vector field→ on TM . S20 S1.3.6 The Ehresmann connection on πTQ : TQ Q associated with an affine connection on Q→.......... S21 ∗ S1.3.7 The Ehresmann connection on πT∗Q : T Q Q associated with an affine connection on Q ..........→ S23 S1.3.8 The Ehresmann connection on πTTQ : TTQ TQ associated with an affine connection on Q ..........→ S23 ∗ S1.3.9 The Ehresmann connection on πT∗TQ : T TQ TQ associated with an affine connection on Q ..........→ S25 ∗ S1.3.10 Representations of ST and ST .................. -
Simply-Riemann-1588263529. Print
Simply Riemann Simply Riemann JEREMY GRAY SIMPLY CHARLY NEW YORK Copyright © 2020 by Jeremy Gray Cover Illustration by José Ramos Cover Design by Scarlett Rugers All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests, write to the publisher at the address below. [email protected] ISBN: 978-1-943657-21-6 Brought to you by http://simplycharly.com Contents Praise for Simply Riemann vii Other Great Lives x Series Editor's Foreword xi Preface xii Introduction 1 1. Riemann's life and times 7 2. Geometry 41 3. Complex functions 64 4. Primes and the zeta function 87 5. Minimal surfaces 97 6. Real functions 108 7. And another thing . 124 8. Riemann's Legacy 126 References 143 Suggested Reading 150 About the Author 152 A Word from the Publisher 153 Praise for Simply Riemann “Jeremy Gray is one of the world’s leading historians of mathematics, and an accomplished author of popular science. In Simply Riemann he combines both talents to give us clear and accessible insights into the astonishing discoveries of Bernhard Riemann—a brilliant but enigmatic mathematician who laid the foundations for several major areas of today’s mathematics, and for Albert Einstein’s General Theory of Relativity.Readable, organized—and simple. Highly recommended.” —Ian Stewart, Emeritus Professor of Mathematics at Warwick University and author of Significant Figures “Very few mathematicians have exercised an influence on the later development of their science comparable to Riemann’s whose work reshaped whole fields and created new ones. -
Complete Connections on Fiber Bundles
Complete connections on fiber bundles Matias del Hoyo IMPA, Rio de Janeiro, Brazil. Abstract Every smooth fiber bundle admits a complete (Ehresmann) connection. This result appears in several references, with a proof on which we have found a gap, that does not seem possible to remedy. In this note we provide a definite proof for this fact, explain the problem with the previous one, and illustrate with examples. We also establish a version of the theorem involving Riemannian submersions. 1 Introduction: A rather tricky exercise An (Ehresmann) connection on a submersion p : E → B is a smooth distribution H ⊂ T E that is complementary to the kernel of the differential, namely T E = H ⊕ ker dp. The distributions H and ker dp are called horizontal and vertical, respectively, and a curve on E is called horizontal (resp. vertical) if its speed only takes values in H (resp. ker dp). Every submersion admits a connection: we can take for instance a Riemannian metric ηE on E and set H as the distribution orthogonal to the fibers. Given p : E → B a submersion and H ⊂ T E a connection, a smooth curve γ : I → B, t0 ∈ I, locally defines a horizontal lift γ˜e : J → E, t0 ∈ J ⊂ I,γ ˜e(t0)= e, for e an arbitrary point in the fiber. This lift is unique if we require J to be maximal, and depends smoothly on e. The connection H is said to be complete if for every γ its horizontal lifts can be defined in the whole domain. In that case, a curve γ induces diffeomorphisms between the fibers by parallel transport.