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Connections on Principal Bundles CONNECTIONS ON PRINCIPAL BUNDLES by Matt Lewis SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE AT DALHOUSIE UNIVERSITY HALIFAX, NOVA SCOTIA DECEMBER 2007 (c) Copyright by Matt Lewis, 2007 Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-39133-4 Our file Notre reference ISBN: 978-0-494-39133-4 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Plntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission. In compliance with the Canadian Conformement a la loi canadienne Privacy Act some supporting sur la protection de la vie privee, forms may have been removed quelques formulaires secondaires from this thesis. ont ete enleves de cette these. While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. Canada DALHOUSIE UNIVERSITY To comply with the Canadian Privacy Act the National Library of Canada has requested that the following pages be removed from this copy of the thesis: Preliminary Pages Examiners Signature Page (pii) Dalhousie Library Copyright Agreement (piii) Appendices Copyright Releases (if applicable) Table of Contents Abstract vi List of Abbreviations and Symbols Used vii Acknowledgements ix Chapter 1 Introduction 1 Chapter 2 Maurer-Cartan Form and Principal Bundles 4 2.1 The Maurer-Cartan Form 4 2.1.1 The Adjoint Action 4 2.1.2 The Frolicher-Nijenhuis Bracket 5 2.1.3 The Maurer-Cartan Form 7 2.1.4 The Darboux Derivative 9 2.2 Fibre Bundles 10 2.2.1 Bundles 10 2.2.2 Principal Bundles 13 2.2.3 Associated Bundles 15 2.2.4 The Frame Bundle . 18 Chapter 3 Connections 20 3.1 Connections 20 3.1.1 Ehresmann Connections 20 3.1.2 Cartan Connections 26 3.1.3 Existence of Cartan Connections 29 3.2 Cartan Geometry 36 Chapter 4 Ehresmann Connections versus Cartan Connections 39 4.1 Parallel Displacement 39 IV 4.1.1 Ehresmann Parallel Displacement 39 4.1.2 Cartan Parallel Displacement 40 4.2 Holonomy 44 4.2.1 Ehresmann Holonomy 44 4.2.2 Cartan Holonomy . 45 4.2.3 Ehresmann Holonomy as Cartan Holonomy 49 4.3 The Covariant Exterior Derivative . 51 Chapter 5 Conclusion 59 Bibliography 62 v Abstract This thesis is devoted to the study of connections in geometry, in particular, on principal bundles. The two types of connections on principal bundles, Ehresmann, and (generalized) Cartan connections, are introduced and the theory of the corresponding geometries are developed. Finally the two connections are compared, for one to see the advantages of having a Cartan connection over a Ehresmann connection. VI List of Abbreviations and Symbols Used Sn the symmetric group on n elements Euc„ Euclidean group acting on Rn O(n) orthogonal group acting on W1 SO(n) special orthogonal group acting on W1 G K H semi-direct product of two groups G and H M,N Smooth, paracompact manifolds G,H,L Lie groups g, f), p Lie algebras to the Lie groups G, H, and L respectively Rh, Lh left and right translations by an element h of a Lie group C(g) the adjoint action of an element g in a group Lie group G cl(G) the closure of a topological group G Ar(M) the set of differential r-forms on M Ar(M, Q) the set of g-valued r-forms on M X^ fundamental vector field corresponding to a vector X € 0 [-,-] ....... Frolicher-Nijenhuis bracket [-, -]8 the commutator on the Lie algebra g U>G Maurer-Cartan form on a Lie group G /* push-forward induced by a diffeomorphism / /* pull-back induced by a diffeomorphism / vii u)f Darboux derivative with respect to a map / : M *• G G >• P v > M right principal G-bundle P[F] associated bundle to P with fibre F 7 Ehresmann connection ijj Cartan connection W generalized Cartan connection $![•] curvature of the corresponding connection (P, 7) Ehresmann geometry (P,u>) Cartan geometry (P,oJ) generalized Cartan geometry r(/\r T*P, V)G set of V-valued pseudotensorial r-forms on P r\or(Ar T*P, V)G the set of I^-valued tensorial r-forms on P 9 soldering form c horizontal lift of a path c c development of a path c dy covariant exterior derivative with respect to 7 d^ covariant exterior derivative with respect to OJ dw covariant exterior derivative with respect to a; vm Acknowledgements I would like to thank my supervisor Roman Smirnov for all his help and encourage­ ment. I would also like to thank my other two readers Ray Mclenaghan and Rob Milson for agreeing to read this thesis. ix Chapter 1 Introduction The history of geometry has gone through three major generalizations from Euclidean geometry. The first generalization came in a series of lectures given by Riemann, at the University of Gottingen in 1854. This "new" manner in regarding geometry, was inevitably entitled Riemannian geometry, and relies entirely on the concept of differentiable manifolds, which is simply a generalization of Euclidean space. The key concept being generalized, is the notion of curvature, where we leave the realm of geometric objects having zero curvature, and proceed in the direction of non-zero curvature, both constant and non-constant. After Riemann, came Klein, who attempted to answer the question "what is ge­ ometry?" , by claiming that "geometry is the study of geometric objects and their in­ variant properties under a given transformation group". He explored this idea in 1872 in what is known as "Klein's Erlangen Program". This development uses the ideas of Riemann, namely differentiable manifolds, and the concept of Lie groups. In the case n of Euclidean geometry, we consider the Euclidean group Eucn = O(n) ix M (or in the case of preserving orientation, the special Euclidean group SE(n) = SO(n) K R"), from which we can form Euclidean space by taking the quotient space Euc„/0(n) (SE(n)/SO(n)). Thus the corresponding symmetry group is given by O(n) (SO(n)). The manner in which Klein effects this generalization, is by considering an arbitrary Lie group G, with a closed subgroup H C G, and look at the space G/H, from which we see the symmetry group will be given by H, that is, we simply generalize the symmetry group from O(n) (SO(n)). The notion of Klein geometry gives rise to the idea of connections on principal bundles, which in this case is simply the Maurer-Cartan form, u>a, on the group G, which satisfies the equation du>o + OJQ AWG = 0 (the structure equation). The final major generalization came from the ideas of Cartan in the early 1920's, who 1 2 joined Klein's ideas and Riemann's. That is, we generalize the symmetry group, and add curvature. The curvature in the sense we speak of here, is the curvature of the the connection on a principal bundle. To study the ideas of Cartan, we need the concept of a Cartan connection, say to, on a principal bundle H »- P v > M. If we compute the value du> + cu AOJ, it will in general be non-zero, in contrast to the value zero for the Maurer-Cartan form. The "curvature" of the connection is given by Q[LO] = dio + u> A u, and can be said to measure the "lumpiness" of the space in question, or simply the failure of the structure equation. The above describes the generalization from Klein geometry to Cartan geometry. The manner in which Cartan geometry generalizes Riemannian geometry, is that the tangent space is generalized. With aid of a Cartan connection on a principal bundle H >• P n > M, we may take the Klein geometry G/H, and "roll" it on the underlying geometry M in question. This connection allows this Klein geometry to roll with "slipping" or "twisting", for an accurate description of M. These three major generalizations can be summarize by the following diagram, adapted from [8] and [9], Euclidean Generalize Klein Geonaetr y Symmetry Group Geonnetr y Add Add Curvature Curvature • • Riema tan tnnian Generalize Car —• =-- Geometry Tangent Space Geometry In the process of defining the Cartan connection, which is what Cartan ultimately wanted, there was the Ehresmann connection, developed by Cartan's student Charles Ehresmann. At first glance, these connections seem unrelated, however if we pass from the original principal bundle, to an associated bundle, we achieve some in­ teresting results.
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