Chapter 2 Bundles
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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. -
Stable Higgs Bundles on Ruled Surfaces 3
STABLE HIGGS BUNDLES ON RULED SURFACES SNEHAJIT MISRA Abstract. Let π : X = PC(E) −→ C be a ruled surface over an algebraically closed field k of characteristic 0, with a fixed polarization L on X. In this paper, we show that pullback of a (semi)stable Higgs bundle on C under π is a L-(semi)stable Higgs bundle. Conversely, if (V,θ) ∗ is a L-(semi)stable Higgs bundle on X with c1(V ) = π (d) for some divisor d of degree d on C and c2(V ) = 0, then there exists a (semi)stable Higgs bundle (W, ψ) of degree d on C whose pullback under π is isomorphic to (V,θ). As a consequence, we get an isomorphism between the corresponding moduli spaces of (semi)stable Higgs bundles. We also show the existence of non-trivial stable Higgs bundle on X whenever g(C) ≥ 2 and the base field is C. 1. Introduction A Higgs bundle on an algebraic variety X is a pair (V, θ) consisting of a vector bundle V 1 over X together with a Higgs field θ : V −→ V ⊗ ΩX such that θ ∧ θ = 0. Higgs bundle comes with a natural stability condition (see Definition 2.3 for stability), which allows one to study the moduli spaces of stable Higgs bundles on X. Higgs bundles on Riemann surfaces were first introduced by Nigel Hitchin in 1987 and subsequently, Simpson extended this notion on higher dimensional varieties. Since then, these objects have been studied by many authors, but very little is known about stability of Higgs bundles on ruled surfaces. -
FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974 FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS 1. Definitions and general examples. 2. Foliations of dimension-one. 3. Higher dimensional foliations; integrability criteria. 4. Foliations of codimension-one; existence theorems. 5. Notions of equivalence; foliated cobordism groups. 6. The general theory; classifying spaces and characteristic classes for foliations. 7. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. 8. Results on closed manifolds; questions of compact leaves and stability. Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject. -
Dirac Operators for Matrix Algebras Converging to Coadjoint Orbits
DIRAC OPERATORS FOR MATRIX ALGEBRAS CONVERGING TO COADJOINT ORBITS MARC A. RIEFFEL Abstract. In the high-energy physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general precise setting for understanding such statements, in which the matrix algebras are viewed as quantum metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. But physicists want even more to treat structures on spheres (and other spaces), such as vector bundles, Yang-Mills functionals, Dirac operators, etc., and they want to approximate these by corresponding structures on matrix algebras. In the present paper we provide a somewhat unified construction of Dirac operators on coadjoint orbits and on the matrix algebras that converge to them. This enables us to prove our main theorem, whose content is that, for the quantum metric-space structures determined by the Dirac operators that we construct, the matrix algebras do indeed converge to the coadjoint orbits, for a quite strong version of quantum Gromov-Hausdorff distance. Contents Introduction 2 1. Ergodic actions and Dirac operators 5 2. Invariance under the group actions 9 3. Charge Conjugation 11 4. Casimir operators 12 5. First facts about spinors 14 6. The C*-metrics 16 7. More about spinors 20 8. Dirac operators for matrix algebras 21 arXiv:2108.01136v1 [math.OA] 2 Aug 2021 9. The fuzzy sphere 23 10. Homogeneous spaces and the Dirac operator for G 29 11. Dirac operators for homogeneous spaces 30 12. Complex structure on coadjoint orbits 36 13. Yet more about spinors 40 14. -
Fluxes, Bundle Gerbes and 2-Hilbert Spaces
Lett Math Phys DOI 10.1007/s11005-017-0971-x Fluxes, bundle gerbes and 2-Hilbert spaces Severin Bunk1 · Richard J. Szabo2 Received: 12 January 2017 / Revised: 13 April 2017 / Accepted: 8 May 2017 © The Author(s) 2017. This article is an open access publication Abstract We elaborate on the construction of a prequantum 2-Hilbert space from a bundle gerbe over a 2-plectic manifold, providing the first steps in a programme of higher geometric quantisation of closed strings in flux compactifications and of M5- branes in C-fields. We review in detail the construction of the 2-category of bundle gerbes and introduce the higher geometrical structures necessary to turn their cate- gories of sections into 2-Hilbert spaces. We work out several explicit examples of 2-Hilbert spaces in the context of closed strings and M5-branes on flat space. We also work out the prequantum 2-Hilbert space associated with an M-theory lift of closed strings described by an asymmetric cyclic orbifold of the SU(2) WZW model, pro- viding an example of sections of a torsion gerbe on a curved background. We describe the dimensional reduction of M-theory to string theory in these settings as a map from 2-isomorphism classes of sections of bundle gerbes to sections of corresponding line bundles, which is compatible with the respective monoidal structures and module actions. Keywords Bundle gerbes · Higher geometric quantisation · Fluxes in string and M-theory · Higher geometric structures Mathematics Subject Classification 81S10 · 53Z05 · 18F99 B Severin Bunk [email protected] Richard J. -
3. Tensor Fields
3.1 The tangent bundle. So far we have considered vectors and tensors at a point. We now wish to consider fields of vectors and tensors. The union of all tangent spaces is called the tangent bundle and denoted TM: TM = ∪x∈M TxM . (3.1.1) The tangent bundle can be given a natural manifold structure derived from the manifold structure of M. Let π : TM → M be the natural projection that associates a vector v ∈ TxM to the point x that it is attached to. Let (UA, ΦA) be an atlas on M. We construct an atlas on TM as follows. The domain of a −1 chart is π (UA) = ∪x∈UA TxM, i.e. it consists of all vectors attached to points that belong to UA. The local coordinates of a vector v are (x1,...,xn, v1,...,vn) where (x1,...,xn) are the coordinates of x and v1,...,vn are the components of the vector with respect to the coordinate basis (as in (2.2.7)). One can easily check that a smooth coordinate trasformation on M induces a smooth coordinate transformation on TM (the transformation of the vector components is given by (2.3.10), so if M is of class Cr, TM is of class Cr−1). In a similar way one defines the cotangent bundle ∗ ∗ T M = ∪x∈M Tx M , (3.1.2) the tensor bundles s s TMr = ∪x∈M TxMr (3.1.3) and the bundle of p-forms p p Λ M = ∪x∈M Λx . (3.1.4) 3.2 Vector and tensor fields. -
Notes on Principal Bundles and Classifying Spaces
Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG]. -
Group Invariant Solutions Without Transversality 2 in Detail, a General Method for Characterizing the Group Invariant Sections of a Given Bundle
GROUP INVARIANT SOLUTIONS WITHOUT TRANSVERSALITY Ian M. Anderson Mark E. Fels Charles G. Torre Department of Mathematics Department of Mathematics Department of Physics Utah State University Utah State University Utah State University Logan, Utah 84322 Logan, Utah 84322 Logan, Utah 84322 Abstract. We present a generalization of Lie’s method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables. The theoretical basis for our method is provided by a general existence theorem for the invariant sections, both local and global, of a bundle on which a finite dimensional Lie group acts. A simple and natural extension of our characterization of invariant sections leads to an intrinsic characterization of the reduced equations for the group invariant solutions for a system of differential equations. The char- acterization of both the invariant sections and the reduced equations are summarized schematically by the kinematic and dynamic reduction diagrams and are illustrated by a number of examples from fluid mechanics, harmonic maps, and general relativity. This work also provides the theoretical foundations for a further detailed study of the reduced equations for group invariant solutions. Keywords. Lie symmetry reduction, group invariant solutions, kinematic reduction diagram, dy- namic reduction diagram. arXiv:math-ph/9910015v2 13 Apr 2000 February , Research supported by NSF grants DMS–9403788 and PHY–9732636 1. Introduction. Lie’s method of symmetry reduction for finding the group invariant solutions to partial differential equations is widely recognized as one of the most general and effective methods for obtaining exact solutions of non-linear partial differential equations. -
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]. -
Composable Geometric Motion Policies Using Multi-Task Pullback Bundle Dynamical Systems
Composable Geometric Motion Policies using Multi-Task Pullback Bundle Dynamical Systems Andrew Bylard, Riccardo Bonalli, Marco Pavone Abstract— Despite decades of work in fast reactive plan- ning and control, challenges remain in developing reactive motion policies on non-Euclidean manifolds and enforcing constraints while avoiding undesirable potential function local minima. This work presents a principled method for designing and fusing desired robot task behaviors into a stable robot motion policy, leveraging the geometric structure of non- Euclidean manifolds, which are prevalent in robot configuration and task spaces. Our Pullback Bundle Dynamical Systems (PBDS) framework drives desired task behaviors and prioritizes tasks using separate position-dependent and position/velocity- dependent Riemannian metrics, respectively, thus simplifying individual task design and modular composition of tasks. For enforcing constraints, we provide a class of metric-based tasks, eliminating local minima by imposing non-conflicting potential functions only for goal region attraction. We also provide a geometric optimization problem for combining tasks inspired by Riemannian Motion Policies (RMPs) that reduces to a simple least-squares problem, and we show that our approach is geometrically well-defined. We demonstrate the 2 PBDS framework on the sphere S and at 300-500 Hz on a manipulator arm, and we provide task design guidance and an open-source Julia library implementation. Overall, this work Fig. 1: Example tree of PBDS task mappings designed to move a ball along presents a fast, easy-to-use framework for generating motion the surface of a sphere to a goal while avoiding obstacles. Depicted are policies without unwanted potential function local minima on manifolds representing: (black) joint configuration for a 7-DoF robot arm general manifolds. -
Differential Topology
Differential Topology: Homework Set # 3. 1) Basic on tangent bundles. Recall that the tangent bundle TM to an extrinsicly defined manifold M k ⊂ Rn is defined to be: n n k n ∼ n TM := {(p, v) ∈ R × R | p ∈ M , v ∈ TpM ⊂ TpR = R } This exercise is designed to give you some practice working with tangent bundles. (i) Show that for a manifold M, the tangent bundle TM is well-defined, i.e. does not depend on the embedding M k ⊂ Rn. (ii) If M, N are a pair of smooth manifolds, show that T (M × N) is diffeomorphic to TM × TN. (iii) If f : M → R is a smooth, positive function, show that the corresponding scaling map φ : TM → TM defined by φ(p, v) = (p, f(p) · v) is a diffeomorphism. (iv) Show that the tangent bundle to S1 (the unit sphere) is diffeomorphic to the cylinder S1 × R. (v) Show that if Sn is an arbitrary sphere, then there exists a diffeomorphism between (TSn) × R and Sn × Rn+1. 2) Relatives of the tangent bundle. Closely associated to the tangent bundle are some other manifolds: (i) For an extrinsicly defined manifold M k ⊂ Rn, define the normal bundle as follows: n n k ⊥ n ∼ n N(M) := {(p, v) ∈ R × R | p ∈ M , v ∈ TpM ⊂ TpR = R } Show that N(M) is an n-dimensional manifold. (ii) Show that for the standard n-sphere Sn ⊂ Rn+1, the normal bundle N(Sn) is diffeomorphic to Sn × R. (iii) Consider the subset T 1M of TM consisting of pairs (p, v) ∈ TM with ||v|| = 1 (where || · || denotes the standard norm on Rn). -
Differential Geometry of Complex Vector Bundles
DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR BUNDLES by Shoshichi Kobayashi This is re-typesetting of the book first published as PUBLICATIONS OF THE MATHEMATICAL SOCIETY OF JAPAN 15 DIFFERENTIAL GEOMETRY OF COMPLEX VECTOR BUNDLES by Shoshichi Kobayashi Kan^oMemorial Lectures 5 Iwanami Shoten, Publishers and Princeton University Press 1987 The present work was typeset by AMS-LATEX, the TEX macro systems of the American Mathematical Society. TEX is the trademark of the American Mathematical Society. ⃝c 2013 by the Mathematical Society of Japan. All rights reserved. The Mathematical Society of Japan retains the copyright of the present work. No part of this work may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copy- right owner. Dedicated to Professor Kentaro Yano It was some 35 years ago that I learned from him Bochner's method of proving vanishing theorems, which plays a central role in this book. Preface In order to construct good moduli spaces for vector bundles over algebraic curves, Mumford introduced the concept of a stable vector bundle. This concept has been generalized to vector bundles and, more generally, coherent sheaves over algebraic manifolds by Takemoto, Bogomolov and Gieseker. As the dif- ferential geometric counterpart to the stability, I introduced the concept of an Einstein{Hermitian vector bundle. The main purpose of this book is to lay a foundation for the theory of Einstein{Hermitian vector bundles. We shall not give a detailed introduction here in this preface since the table of contents is fairly self-explanatory and, furthermore, each chapter is headed by a brief introduction.