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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. -
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 Comparison of Differential Calculus and Differential Geometry in Two
1 A Two-Dimensional Comparison of Differential Calculus and Differential Geometry Andrew Grossfield, Ph.D Vaughn College of Aeronautics and Technology Abstract and Introduction: Plane geometry is mainly the study of the properties of polygons and circles. Differential geometry is the study of curves that can be locally approximated by straight line segments. Differential calculus is the study of functions. These functions of calculus can be viewed as single-valued branches of curves in a coordinate system where the horizontal variable controls the vertical variable. In both studies the derivative multiplies incremental changes in the horizontal variable to yield incremental changes in the vertical variable and both studies possess the same rules of differentiation and integration. It seems that the two studies should be identical, that is, isomorphic. And, yet, students should be aware of important differences. In differential geometry, the horizontal and vertical units have the same dimensional units. In differential calculus the horizontal and vertical units are usually different, e.g., height vs. time. There are differences in the two studies with respect to the distance between points. In differential geometry, the Pythagorean slant distance formula prevails, while in the 2- dimensional plane of differential calculus there is no concept of slant distance. The derivative has a different meaning in each of the two subjects. In differential geometry, the slope of the tangent line determines the direction of the tangent line; that is, the angle with the horizontal axis. In differential calculus, there is no concept of direction; instead, the derivative describes a rate of change. In differential geometry the line described by the equation y = x subtends an angle, α, of 45° with the horizontal, but in calculus the linear relation, h = t, bears no concept of direction. -
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 .................. -
Riemann's Contribution to Differential Geometry
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematics 9 (1982) l-18 RIEMANN'S CONTRIBUTION TO DIFFERENTIAL GEOMETRY BY ESTHER PORTNOY UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN, URBANA, IL 61801 SUMMARIES In order to make a reasonable assessment of the significance of Riemann's role in the history of dif- ferential geometry, not unduly influenced by his rep- utation as a great mathematician, we must examine the contents of his geometric writings and consider the response of other mathematicians in the years immedi- ately following their publication. Pour juger adkquatement le role de Riemann dans le developpement de la geometric differentielle sans etre influence outre mesure par sa reputation de trks grand mathematicien, nous devons &udier le contenu de ses travaux en geometric et prendre en consideration les reactions des autres mathematiciens au tours de trois an&es qui suivirent leur publication. Urn Riemann's Einfluss auf die Entwicklung der Differentialgeometrie richtig einzuschZtzen, ohne sich von seinem Ruf als bedeutender Mathematiker iiberm;issig beeindrucken zu lassen, ist es notwendig den Inhalt seiner geometrischen Schriften und die Haltung zeitgen&sischer Mathematiker unmittelbar nach ihrer Verijffentlichung zu untersuchen. On June 10, 1854, Georg Friedrich Bernhard Riemann read his probationary lecture, "iber die Hypothesen welche der Geometrie zu Grunde liegen," before the Philosophical Faculty at Gdttingen ill. His biographer, Dedekind [1892, 5491, reported that Riemann had worked hard to make the lecture understandable to nonmathematicians in the audience, and that the result was a masterpiece of presentation, in which the ideas were set forth clearly without the aid of analytic techniques. -
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. -
Linearization Via the Lie Derivative ∗
Electron. J. Diff. Eqns., Monograph 02, 2000 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp) Linearization via the Lie Derivative ∗ Carmen Chicone & Richard Swanson Abstract The standard proof of the Grobman–Hartman linearization theorem for a flow at a hyperbolic rest point proceeds by first establishing the analogous result for hyperbolic fixed points of local diffeomorphisms. In this exposition we present a simple direct proof that avoids the discrete case altogether. We give new proofs for Hartman’s smoothness results: A 2 flow is 1 linearizable at a hyperbolic sink, and a 2 flow in the C C C plane is 1 linearizable at a hyperbolic rest point. Also, we formulate C and prove some new results on smooth linearization for special classes of quasi-linear vector fields where either the nonlinear part is restricted or additional conditions on the spectrum of the linear part (not related to resonance conditions) are imposed. Contents 1 Introduction 2 2 Continuous Conjugacy 4 3 Smooth Conjugacy 7 3.1 Hyperbolic Sinks . 10 3.1.1 Smooth Linearization on the Line . 32 3.2 Hyperbolic Saddles . 34 4 Linearization of Special Vector Fields 45 4.1 Special Vector Fields . 46 4.2 Saddles . 50 4.3 Infinitesimal Conjugacy and Fiber Contractions . 50 4.4 Sources and Sinks . 51 ∗Mathematics Subject Classifications: 34-02, 34C20, 37D05, 37G10. Key words: Smooth linearization, Lie derivative, Hartman, Grobman, hyperbolic rest point, fiber contraction, Dorroh smoothing. c 2000 Southwest Texas State University. Submitted November 14, 2000. -
Smoothing Maps Into Algebraic Sets and Spaces of Flat Connections
SMOOTHING MAPS INTO ALGEBRAIC SETS AND SPACES OF FLAT CONNECTIONS THOMAS BAIRD AND DANIEL A. RAMRAS n Abstract. Let X ⊂ R be a real algebraic set and M a smooth, closed manifold. We show that all continuous maps M ! X are homotopic (in X) to C1 maps. We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations, and we establish lower bounds on the ranks of the homotopy groups of spaces of flat connections over aspherical manifolds. 1. Introduction The first goal of this paper is to prove the following result about the differential topology of algebraic sets. Theorem 1.1 (Section2) . Let X ⊂ Rn be a (possibly singular) real algebraic set, and let f : M ! X be a continuous map from a smooth, closed manifold M. Then there exists a map g : M ! X, and a homotopy H : M × I ! X connecting f and g g, such that the composite M ! X,! Rn is C1. The problem of smoothing maps into algebraic sets seems natural, but we have not found mention of it in the literature. We consulted several experts in real algebraic geometry; some expected our result to hold, and some did not. Our proof proceeds by embedding X as the singular set of an irreducible, quasi- projective variety Y and using a resolution of singularities Ye ! Y for which the inverse image of X is a divisor with normal crossing singularities. Basic facts about neighborhoods of algebraic sets then reduce the problem to the case of normal crossing divisors, which can be handled by differential-geometric means. -
Hamilton's Ricci Flow
The University of Melbourne, Department of Mathematics and Statistics Hamilton's Ricci Flow Nick Sheridan Supervisor: Associate Professor Craig Hodgson Second Reader: Professor Hyam Rubinstein Honours Thesis, November 2006. Abstract The aim of this project is to introduce the basics of Hamilton's Ricci Flow. The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it \rounder", in the hope that one may draw topological conclusions from the existence of such \round" metrics. Indeed, the Ricci flow has recently been used to prove two very deep theorems in topology, namely the Geometrization and Poincar´eConjectures. We begin with a brief survey of the differential geometry that is needed in the Ricci flow, then proceed to introduce its basic properties and the basic techniques used to understand it, for example, proving existence and uniqueness and bounds on derivatives of curvature under the Ricci flow using the maximum principle. We use these results to prove the \original" Ricci flow theorem { the 1982 theorem of Richard Hamilton that closed 3-manifolds which admit metrics of strictly positive Ricci curvature are diffeomorphic to quotients of the round 3-sphere by finite groups of isometries acting freely. We conclude with a qualitative discussion of the ideas behind the proof of the Geometrization Conjecture using the Ricci flow. Most of the project is based on the book by Chow and Knopf [6], the notes by Peter Topping [28] (which have recently been made into a book, see [29]), the papers of Richard Hamilton (in particular [9]) and the lecture course on Geometric Evolution Equations presented by Ben Andrews at the 2006 ICE-EM Graduate School held at the University of Queensland. -
WHAT IS a CONNECTION, and WHAT IS IT GOOD FOR? Contents 1. Introduction 2 2. the Search for a Good Directional Derivative 3 3. F
WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR? TIMOTHY E. GOLDBERG Abstract. In the study of differentiable manifolds, there are several different objects that go by the name of \connection". I will describe some of these objects, and show how they are related to each other. The motivation for many notions of a connection is the search for a sufficiently nice directional derivative, and this will be my starting point as well. The story will by necessity include many supporting characters from differential geometry, all of whom will receive a brief but hopefully sufficient introduction. I apologize for my ungrammatical title. Contents 1. Introduction 2 2. The search for a good directional derivative 3 3. Fiber bundles and Ehresmann connections 7 4. A quick word about curvature 10 5. Principal bundles and principal bundle connections 11 6. Associated bundles 14 7. Vector bundles and Koszul connections 15 8. The tangent bundle 18 References 19 Date: 26 March 2008. 1 1. Introduction In the study of differentiable manifolds, there are several different objects that go by the name of \connection", and this has been confusing me for some time now. One solution to this dilemma was to promise myself that I would some day present a talk about connections in the Olivetti Club at Cornell University. That day has come, and this document contains my notes for this talk. In the interests of brevity, I do not include too many technical details, and instead refer the reader to some lovely references. My main references were [2], [4], and [5]. -
On Lie Derivation of Spinors Against Arbitrary Tangent Vector Fields
On Lie derivation of spinors against arbitrary tangent vector fields Andras´ LASZL´ O´ [email protected] Wigner RCP, Budapest, Hungary (joint work with L.Andersson and I.Rácz) CERS8 Workshop Brno, 17th February 2018 On Lie derivation of spinors against arbitrary tangent vector fields – p. 1 Preliminaries Ordinary Lie derivation. Take a one-parameter group (φt)t∈R of diffeomorphisms over a manifold M. Lie derivation against that is defined on the smooth sections χ of the mixed tensor algebra of T (M) and T ∗(M), with the formula: L φ∗ χ := ∂t φ∗ −t χ t=0 It has explicit formula: on T (M) : Lu(χ)=[u,χ] , a on F (M) := M× R : Lu(χ)= u da (χ) , ∗ a a on T (M) : Lu(χ)= u da (χ) + d(u χa), where u is the unique tangent vector field underlying (φt)t∈R. The map u 7→ Lu is faithful Lie algebra representation. On Lie derivation of spinors against arbitrary tangent vector fields – p. 2 Lie derivation on a vector bundle. Take a vector bundle V (M) over M. Take a one-parameter group of diffeomorphisms of the total space of V (M), which preserves the vector bundle structure. The ∂t() against the flows of these are vector bundle Lie derivations. t=0 (I.Kolár,P.Michor,K.Slovák:ˇ Natural operations in differential geometry; Springer 1993) Explicit expression: take a preferred covariant derivation ∇, then over the sections χ of V (M) one can express these as ∇ A a A A B L χ = u ∇a χ − CB χ (u,C) That is: Lie derivations on a vector bundle are uniquely characterized by their horizontal a A part u ∇a and vertical part CB .