DOCSLIB.ORG

Foliations with Transversal Quaternionic Structures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Foliations with Transversal Quaternionic Structures The Erwin Schrodinger International Boltzmanngasse ESI Institute for Mathematical Physics A Wien Austria Foliations with Transversal Quaternionic Structures Paolo Piccinni Izu Vaisman Vienna Preprint ESI January Supp orted by Federal Ministry of Science and Transp ort Austria Available via httpwwwesiacat Foliatio ns with Transversal Quaternionic Structures by Paolo Piccinni and Izu Vaisman ABSTRACT We consider manifolds equipp ed with a foliation F of co dimen sion q and an almost quaternionic structure Q on the transversal bundle of F After discussing conditions of pro jectability and integrability of Q we study the transversal twistor space Z F which by denition consists of the Qcompatible almost complex structures We show that Z F can b e endowed with a lifted b foliation F and two natural almost complex structures J J on the transversal 1 2 b bundle of F We establish the conditions which ensure the pro jectability of J 1 and J and the integrability of J J is never integrable 2 1 2 Preliminaries We recall the basic denitions of quaternionic geometry eg The general framework is the C category An almost hypercomplex structure on a q dimensional dierentiable man 4q ifold N is an ordered triple H I I I of almost complex structures 1 2 3 satisfying the quaternionic identities I I I for and cyclic p ermutations If the structures I I I are integrable H is said to b e 1 2 3 a hypercomplex structure Mathematics Subject Classication C C Key words and phrases FoliationQuaternionic Structure Twistor Space Acknow ledgements This pap er was started while the second author visited the Istituto di Matematica Universita di Roma under the auspices of the CNR Italy He is ex pressing here his thanks to the hosts The rst author acknowledges supp ort by MURST of Italy and supp ort and hospitality by the Erwin Schrodinger Institute for Mathematical Physics in Vienna If H I I I is an almost hyp ercomplex structure any triple J J 1 2 3 1 2 J obtained from I I I by multiplying by a matrix of SO is again 3 1 2 3 an almost hyp ercomplex structure Moreover there exists a set of compati ble almost complex structures asso ciated with a given almost hyp ercomplex structure namely the set of all J a I a I a I where a a a are 1 1 2 2 3 3 1 2 3 2 2 2 functions satisfying a a a 1 2 3 4q An almost quaternionic structure on the manifold N is a rank vec tor subbundle Q of the endomorphism bundle E ndTN lo cally spanned by almost hyp ercomplex structures H I I I which are related by SO 1 2 3 matrices on the intersections of trivializing op en sets A quaternionic struc 4q ture on the manifold N is an almost quaternionic structure such that there exists a torsionless connection r of TN which when extended to the vector bundle E ndTN preserves the subbundle Q ie rQ Q The existence of the Qpreserving torsionless connection r is not equivalent with the inte grability of Q as a Gstructure The existence of a at torsionless connection which preserves Q implies that Q can b e obtained from lo cal quaternionic 4q co ordinates on N If an almost quaternionic structure Q is xed on N the local bases I I I which span the vector bundle Q are also called local com 1 2 3 patible almost hypercomplex structures and any lo cal J a I a I a I 1 1 2 2 3 3 2 2 2 is called a local Qcompatible almost complex structure a a with a 3 2 1 A Riemannian metric g on a almost hyp ercomplex manifold N H is almost hyperhermitian resp ectively almost hyperkahler if it is almost Hermitian resp ectively almost Kahlerian with resp ect to all the structures I of H Then g also is compatible with any H compatible structure J Similarly on an almost quaternionic manifold N Q the met ric g is quaternion Hermitian if it is Hermitian with resp ect to the lo cal bases I of Q and it is quaternion Kahler if it is quaternion Hermitian and Q is parallel ie rQ Q with resp ect to the LeviCivita connection r of g In b oth cases the prop erty says nothing ab out the integrability of the struc tures I The terms are also used for manifolds endowed with the resp ective structures Of course a hyp erkahler manifold necessarily is hyp ercomplex and a quaternion Kahler manifold necessarily is quaternionic The twistor space ZN of an almost quaternionic manifold N Q is de ned as the manifold of the Qcompatible almost complex structures of the 2 tangent spaces of N Thus ZN is an S bundle asso ciated with the vector bundle Q where Q has the metric which makes the lo cal bases H I I I 1 2 3 orthonormal bases p+4q Now let us consider a C manifold M equipp ed with a pdimensional foliation F Denote by L T F the tangent bundle of F and by F T M L its transversal vector bundle of rank q We will often identify the transversal bundle F with a complementary distribution E of L ie a split ting of the exact sequence F L T F TM Almost hyp ercomplex and almost quaternionic structures can b e dened similarly on vector bundles of rank q Accordingly they will b e reductions of the structure group of the bundle to G where the group G GLq H for the almost hyp ercomplex structures and GLq H S p G GLq H S p I d for the almost quaternionic structures H is the algebra of the quaternions Furthermore almost hyp erhermitian and quaternion Hermitian structures corresp ond to the structure groups S pq and S pq S p resp ectively We will consider structures of these typ es on the transversal bundle F of a foliation F and refer to them as transversal almost hypercomplex transver sal almost quaternionic etc structures of the foliation F Such structures sp oradically app eared in the literature eg In what follows we use Bott connections D TM F F A Bott connection is a connection on the transversal bundle F which extends the partial connection D L F F given by D s Y X s Y where Y is a tangent vector eld of the leaves of the foliation F and X s is any vector eld on M such that X s s F always denotes s spaces of global cross sections of vector bundles Notice that an identication F E where TM E L implies the replacement of by D X Y X X E Y b eing the pro jection TM E A Riemannian metric g splits TM T F T F and we will take E T F F Then in particular D can b e extended to a Bott connection D by dening D r X E where r is the LeviCivita connection of X X g For Riemannian foliations this Bott connection D is the unique torsionless metric connection of the normal bundle T F F Denition a A transversal almost hyp ercomplex structure H I I I of F is pro jectable if the partial connection D preserves the struc 1 2 3 tures I i e DI b A transversal almost quaternionic structure Q E nd F is pro jectable if D preserves Q DQ Q The pro jectability condition of Q can b e formulated in terms of lo cal bases H I I I Namely Q is pro jectable i for any choice of a lo cal 1 2 3 basis H I I I there exist lo cal forms such that 1 2 3 DI I I DI I I DI I I 1 2 3 2 1 3 3 1 2 As a matter of fact if equations of the typ e hold for some choice of H similar equations hold for any choice of H If J E nd F we may also see it as a cross section of E nd E and for Y T F s F we have D J s D J s J D s Y J X J Y X Y Y Y s s for any X TM such that s X A cross section s is projectable if s s s pro jects to a tangent vector eld of any lo cal space of slices of the foliation F From it follows that J is pro jectable i J s is pro jectable whenever s is pro jectable Therefore pro jectability in the sense of Denition a means that we have a structure which is the lift of almost hyp ercomplex structures of the lo cal slice spaces The same is true in the case of Denition b see Prop osition later on Accordingly we will give Denition A pro jectable transversal almost hyp ercomplex or almost quaternionic structure of a foliation F is integrable if the pro jected structures of the lo cal slice spaces are hyp ercomplex or quaternionic resp ectively If integrability holds the word almost will b e omitted Examples We b egin by an example of a foliation with a transversal almost hyp er complex structure Let F b e a transversally holomorphic foliation of real p+2q co dimension q on a manifold N This means that on N there are lo cal u a a u a co ordinates y z z where y are real co ordinates and z are complex co ordinates with holomorphic transition functions such that F is dened by a a z const z const Furthermore assume that g is a bundlelike Rie mannian metric on N which is F transversally Kahler ie a b a b g g z zdz dz g z zdz dz ab ab where the rst two terms dene a Kahler metric in the co ordinates z and u the remaining unexplicited terms contain dy In this pap er we use the Einstein summation convention Now consider the manifold M dened by the total space of the conormal bundle of the foliation F ie the annihilator of the tangent bundle T F If E is the g orthogonal bundle of F then M E On M there exists a natural lift F of F such that the leaves of F are covering spaces of the leaves of F Moreover the lo cal slice spaces of F are cotangent bundles of Kahler manifolds It is not dicult to construct an almost hyp ercomplex structure on such a cotangent bundle If we do this on the lo cal slice spaces the obtained structures glue up to a pro jectable almost hyp ercomplex structure transversal to F The construction of the almost hyp ercomplex structure of the cotangent bundle of
Load more

Recommended publications
  • 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].
    [Show full text]
  • Principal Bundles, Vector Bundles and Connections
    Appendix A Principal Bundles, Vector Bundles and Connections Abstract The appendix defines fiber bundles, principal bundles and their associate vector bundles, recall the definitions of frame bundles, the orthonormal frame bun- dle, jet bundles, product bundles and the Whitney sums of bundles. Next, equivalent definitions of connections in principal bundles and in their associate vector bundles are presented and it is shown how these concepts are related to the concept of a covariant derivative in the base manifold of the bundle. Also, the concept of exterior covariant derivatives (crucial for the formulation of gauge theories) and the meaning of a curvature and torsion of a linear connection in a manifold is recalled. The concept of covariant derivative in vector bundles is also analyzed in details in a way which, in particular, is necessary for the presentation of the theory in Chap. 12. Propositions are in general presented without proofs, which can be found, e.g., in Choquet-Bruhat et al. (Analysis, Manifolds and Physics. North-Holland, Amsterdam, 1982), Frankel (The Geometry of Physics. Cambridge University Press, Cambridge, 1997), Kobayashi and Nomizu (Foundations of Differential Geometry. Interscience Publishers, New York, 1963), Naber (Topology, Geometry and Gauge Fields. Interactions. Applied Mathematical Sciences. Springer, New York, 2000), Nash and Sen (Topology and Geometry for Physicists. Academic, London, 1983), Nicolescu (Notes on Seiberg-Witten Theory. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2000), Osborn (Vector Bundles. Academic, New York, 1982), and Palais (The Geometrization of Physics. Lecture Notes from a Course at the National Tsing Hua University, Hsinchu, 1981). A.1 Fiber Bundles Definition A.1 A fiber bundle over M with Lie group G will be denoted by E D .E; M;;G; F/.
    [Show full text]
  • On Fibre Bundles and Differential Geometry
    Lectures On Fibre Bundles and Differential Geometry By J.L. Koszul Notes by S. Ramanan No part of this book may be reproduced in any form by print, microfilm or any other means without written permission from the Tata Insti- tute of Fundamental Research, Apollo Pier Road, Bombay-1 Tata Institute of Fundamental Research, Bombay 1960 Contents 1 Differential Calculus 1 1.1 .............................. 1 1.2 Derivationlaws ...................... 2 1.3 Derivation laws in associated modules . 4 1.4 TheLiederivative... ..... ...... ..... .. 6 1.5 Lie derivation and exterior product . 8 1.6 Exterior differentiation . 8 1.7 Explicit formula for exterior differentiation . 10 1.8 Exterior differentiation and exterior product . 11 1.9 The curvature form . 12 1.10 Relations between different derivation laws . 16 1.11 Derivation law in C .................... 17 1.12 Connections in pseudo-Riemannian manifolds . 18 1.13 Formulae in local coordinates . 19 2 Differentiable Bundles 21 2.1 .............................. 21 2.2 Homomorphisms of bundles . 22 2.3 Trivial bundles . 23 2.4 Induced bundles . 24 2.5 Examples ......................... 25 2.6 Associated bundles . 26 2.7 Vector fields on manifolds . 29 2.8 Vector fields on differentiable principal bundles . 30 2.9 Projections vector fields . 31 iii iv Contents 3 Connections on Principal Bundles 35 3.1 .............................. 35 3.2 Horizontal vector fields . 37 3.3 Connection form . 39 3.4 Connection on Induced bundles . 39 3.5 Maurer-Cartan equations . 40 3.6 Curvatureforms. 43 3.7 Examples ......................... 46 4 Holonomy Groups 49 4.1 Integral paths . 49 4.2 Displacement along paths . 50 4.3 Holonomygroup ....................
    [Show full text]
  • Foliated Manifolds, Algebraic K-Theory, and a Secondary Invariant
    Foliated manifolds, algebraic K-theory, and a secondary invariant Ulrich Bunke∗ June 25, 2018 Abstract We introduce a C=Z-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential K-theory, or alternatively, in terms of η-invariants of Dirac operators and local correction terms. Initially, the construction of the element in C=Z involves additional choices. But if the codimension of the foliation is sufficiently small, then this element is independent of these choices and therefore an invariant of the data listed above. We show that the invariant comprises various classical invariants like Adams' e- invariant, the ρ-invariant of twisted Dirac operators, or the Godbillon-Vey invariant from foliation theory. Using methods from differential cohomology theory we construct a regulator map from the algebraic K-theory of smooth functions on a manifold to its connective K-theory with C=Z coefficients. Our main result is a formula for the invariant in terms of this regulator and integration in algebraic and topological K-theory. arXiv:1507.06404v4 [math.KT] 22 Jun 2018 Contents 1 Introduction 2 2 Basic notions 4 2.1 Foliated manifolds . .4 2.2 Filtrations on the de Rham complex . .6 2.3 Vector bundles with flat partial connections . .8 2.4 Connections and characteristic forms . 10 2.5 Characteristic forms of real foliations . 12 2.6 Transgression . 14 ∗NWF I - Mathematik, Universit¨at Regensburg, 93040 Regensburg, GERMANY, [email protected] 1 3 Differential K-theory 18 3.1 Basic structures .
    [Show full text]
  • CHAPTER 4: PRINCIPAL BUNDLES 4.1 Lie Groups a Lie Group Is A
    CHAPTER 4: PRINCIPAL BUNDLES 4.1 Lie groups A Lie group is a group G which is also a smooth manifold such that the mul- tiplication map G × G → G, (a, b) 7→ ab, and the inverse G → G, a 7→ a−1, are smooth. Actually, one can prove (but this is not easy) that it is sufficient to assume continuety, smoothness comes free. (This was one of the famous problems listed by David Hilbert in his address to the international congress of mathematicians in 1900. The result was proven by D. Montgomery and L. Zippin in 1952.) Examples The vector space Rn is a Lie group. The group multiplication is just the addition of vectors. The set GL(n, R) of invertible real n × n matrices is a Lie group with respect to the usual matrix multiplication. Theorem. Any closed subgroup of a Lie group is a Lie group. The proof is complicated. See for example S. Helgason: Differential Geometry, Lie Groups and Symmetric Spaces, section II.2. The theorem gives an additional set of examples of Lie groups: The group of real orthogonal matrices, the group of complex unitary matrices, the group of invertible upper triangular matrices .... For a fixed a ∈ G in a Lie group we can define a pair of smooth maps, the left translation la : G → G, la(g) = ag, and the right translation ra : G → G, ra(g) = ga. 1 We say that a vector field X ∈ D (G) is left (resp. right) invariant if (la)∗X = X (resp. (ra)∗X = X) for all a ∈ G.
    [Show full text]
  • Group-Valued Momentum Maps for Actions of Automorphism Groups
    Group-valued momentum maps for actions of automorphism groups Tobias Dieza and Tudor S. Ratiub February 5, 2020 The space of smooth sections of a symplectic fiber bundle carries a natural symplectic structure. We provide a general framework to determine the momentum map for the action of the group of bundle automorphism on this space. Since, in general, this action does not admit a classical momentum map, we introduce the more general class of group-valued momentum maps which is inspired by the Poisson Lie setting. In this approach, the group-valued momentum map assigns to every section of the symplectic fiber bundle a principal circle-bundle with connection. The power of this general framework is illustrated in many examples: we construct generalized Clebsch variables for fluids with integral helicity; the anti-canonical bundle turns out to be the momentum map for the action of the group of symplectomorphisms on the space of compatible complex structures; the Teichmüller moduli space is realized as a symplectic orbit reduced space associated to a coadjoint orbit of SL 2; R and spaces related to the other coadjoint orbits are identified and studied.¹ º Moreover, we show that the momentum map for the group of bundle automorphisms on the space of connections over a Riemann surface encodes, besides the curvature, also topological information of the bundle. Keywords: symplectic structure on moduli spaces of geometric structures, infinite- dimensional symplectic geometry, momentum maps, symplectic fiber bundles, differential characters, Kähler geometry, Teichmüller space MSC 2010: 53D20, (58D27, 53C08, 53C10, 58B99, 32G15) a Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany and Institut für Theoretische Physik, Universität Leipzig, 04009 Leipzig, Germany and Institute of Ap- arXiv:2002.01273v1 [math.DG] 4 Feb 2020 plied Mathematics, Delft University of Technology, 2628 XE Delft, Netherlands.
    [Show full text]
  • A Foliated Seiberg-Witten Theory a Dissertation Presented to The
    A Foliated Seiberg-Witten Theory A Dissertation presented to the Faculty of the Graduate School University of Missouri In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by ANDREW RENNER Dr. Shuguang Wang, Dissertation Supervisor MAY 2016 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled A Foliated Seiberg-Witten Theory presented by Andrew Renner, a candidate for the degree of Doctor of Philosophy of Mathematics, and hereby certify that in their opinion it is worthy of acceptance. Professor Shuguang Wang Professor Zhenbo Qin Professor Jan Segert Professor Jianguo Sun ACKNOWLEDGEMENTS I'd like to thank my advisor Dr. Shuguang Wang for his support and guidance throughout my graduate studies here at the University of Missouri. I would further like to thank my committee members Dr. Jan Segert, Dr. Zhenbo Qin, and Dr. Jianguo Sun for their support and encouragement. Finally, I'd like to thank all of the other members of the MU mathematics department faculty, staff and students, they have all made my time here memorable and enjoyable. ii TABLE OF CONTENTS Acknowledgements ii Abstract v 1 Introduction 1 2 Foliations 7 2.1 Transverse Geometry . 10 2.2 The Transverse Riemannian Connection . 16 3 Foliated Dirac Bundles 24 3.1 Foliated Principal Bundles . 24 3.2 Dirac Bundles . 27 3.3 Associated Connections . 29 3.4 Second Order Differential Operators . 33 3.5 Global Analysis . 36 3.6 Weitzenb¨ock Identities . 40 4 Elliptic Theory 47 4.1 Sobolev Spaces . 47 4.1.1 Classical Theory . 47 iii 4.1.2 Basic Sobolev Theory .
    [Show full text]
  • Orientability of Vector Bundles Over Real Flag Manifolds
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 159 (2012) 2774–2786 Contents lists available at SciVerse ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Orientability of vector bundles over real flag manifolds ✩ ∗ Mauro Patrão a, Luiz A.B. San Martin b, ,1, Laércio J. dos Santos c, Lucas Seco a a Department of Mathematics, Universidade de Brasília, Brasília – DF, Brazil b Institute of Mathematics, Universidade Estadual de Campinas, Campinas – SP, Brazil c Department of Mathematics, Universidade Federal de Juiz de Fora – MG, Brazil article info abstract Article history: We investigate the orientability of a class of vector bundles over flag manifolds of real Received 27 June 2011 semi-simple Lie groups, which include the tangent bundle and also stable bundles of Accepted 5 March 2012 certain gradient flows. Closed formulas, in terms of roots, are provided. © 2012 Elsevier B.V. All rights reserved. MSC: 57R22 14M15 Keywords: Vector bundles Orientability Flag manifold 1. Introduction We investigate the orientability of a class of vector bundles over flag manifolds of real semi-simple Lie groups, the so- called (generalized) real flag manifolds. These include the tangent bundle and also stable bundles of some gradient flows on these manifolds which were considered elsewhere (see Section 3 of Duistermat, Kolk and Varadarajan [7] and Section 5 of the present article). We get closed formulas, in terms of roots associated to the real flag manifolds, to decide when they are orientable. As far as we know, our results and methods of proof are not known.
    [Show full text]
  • Gauge Theory Summer Term 2009 OTSDAM P EOMETRIE in G
    Christian B¨ar Gauge Theory Summer Term 2009 OTSDAM P EOMETRIE IN G Version of July 26, 2011 c Christian B¨ar 2011. All rights reserved. Picture on the title page is taken from www.sxc.hu Contents Preface v 1 Lie groups and Lie algebras 1 1.1 Liegroups.................................... 1 1.2 Liealgebras................................... 4 1.3 Representations................................. 7 1.4 Theexponentialmap.............................. 15 1.5 Groupactions.................................. 21 2 Bundle theory 31 2.1 Fiberbundles.................................. 31 2.2 Principalbundles................................ 35 2.3 Connections................................... 44 2.4 Curvature.................................... 49 2.5 Characteristicclasses. 53 2.6 Paralleltransport............................... 59 2.7 Gaugetransformations. 69 3 Applications to Physics 73 3.1 TheHodge-staroperator. 73 3.2 Electrodynamics ................................ 78 3.3 Yang-Millsfields ................................ 91 4 Algebraic Topology 101 4.1 Homotopytheory................................101 4.2 Homologytheory ................................110 4.3 Orientations and the fundamental class . 120 5 4-dimensional Manifolds 125 5.1 Theintersectionform ............................. 125 5.2 Classificationresults . 132 5.3 Donaldson’stheorem .............................. 138 iii Preface These are the lecture notes of an introductory course on gauge theory which I taught at Potsdam University in 2009. The aim was to develop the mathematical underpinnings
    [Show full text]
  • Lecture 1 This Lecture Gives the Essential Definitions of Fiber And
    Lecture 1 This lecture gives the essential definitions of fiber and principal bun- dles as well as the construction of associated bundles. Any propositions will have relatively easy proofs which will be left as exercises. We start with the standard definition of a fiber bundle. All manifolds will be smooth. Definition 1. A fiber bundle of manifolds consists of manifolds F , E and B along with a map π : E → B such that for every b ∈ B there −1 exists a neighborhood Ub and a diffeomorphism φb : Ub × F → π (Ub) such that proj1 = π ◦φb where proj1 is the projection to the first factor. Examples include vector bundles, principal bundles and, by Ehres- mann’s Theorem, any smooth submersion with π proper. Exercise 1.1. Give an example to show that if π is a submersion but is not proper, then π : E → B is not necessarily a fiber bundle. Now we set up some general notation that will be in force through- out this lecture. Let G be a topological group and C a category whose objects are CW complexes plus extra structure. I.e. we take C to be a category with a faithful functor from C to the category of CW complexes and continuous maps. A G-representation in C is an ob- ject A of C and a continuous group homomorphism ψ : G → Aut(A) (here Aut(A) will be given the subspace topology of the compact open topology on End(A)). With this in mind, we define: Definition 2. i) A G-bundle with fiber A in C is a fiber bundle π : E → B with a right G action and with fiber F ≈ A as a G-space.
    [Show full text]
  • CONNECTIONS on PRINCIPAL FIBRE BUNDLES 1. Introduction
    CONNECTIONS ON PRINCIPAL FIBRE BUNDLES SCOTT MORRISON Abstract. Complete definitions and motivations are given for principal fibre bundles, connections on these, and associated vec- tor bundles. The relationships between various concepts of differ- entiation in these settings are proved. Finally, we briefly give an outline of the theory of curvature in terms of connections. 1. Introduction The motivation for defining principal fibre bundles and connections on them comes from the idea of parallel transport and covariant dif- ferentiation on a smooth manifold. Defining a derivative of one vector field with respect to another requires a rule for comparing vectors in nearby tangent spaces. The tangent space to each point of a mani- fold looks the same, that is, each is isomorphic to Rn, but there is no canonical isomorphism between these tangent spaces. As such, there is no way to differentiate a vector field along a curve using the usual idea of the limit of a difference quotient, because the vectors being com- pared live in different tangent spaces. The solution to this problem is to define a rule for parallel transport on the tangent bundle (as a set, the collection of tangent spaces), which, given a smooth curve joining two points on a manifold, produces an isomorphism of the correspond- ing tangent spaces. This then enables vector fields to be differentiated along curves, and we think of the covariant derivative of one vector field with respect to another as the derivative of the first along an integral curve of the second. We can further note that parallel transport can not only be thought of as carrying a single vector from one tangent space to another, but as carrying a frame for one to a frame for the other.
    [Show full text]
  • Connections on Principal Bundles
    Connections on principal bundles Santiago Quintero de los Ríos* December 16, 2020 Contents 1 Connections on principal bundles 1 1.1 Connections as horizontal distributions ........................ 1 1.2 Connections as 1-forms ................................ 3 1.3 Local expressions, or, why physicists did nothing wrong ............... 5 1.4 Horizontal lifts, parallel transport and holonomy ................... 8 2 Curvature 11 2.1 The curvature 2-form and structure equation ..................... 11 2.2 Local expressions (Curvature Edition) ......................... 14 2.3 The exterior covariant derivative ........................... 15 3 The relation with connections on vector bundles 17 3.1 From vector bundles to principal bundles ....................... 17 3.2 Interlude: Associated bundles ............................. 19 3.3 From principal bundles to vector bundles ....................... 22 3.4 In physics language .................................. 24 Notation These notes compile some general facts about connections on principal bundles, and their relation to connections on vector bundles. A few notes on notation: Weare working in the context of a principal 퐺-bundle 푃 over a manifold 휋 푀. This we denote as 퐺 ↪ 푃 → 푀; where 휋 ∶ 푃 → 푀 is the projection map. The right action of 퐺 on 푃 is denoted as 휎 ∶ 푃 ×퐺 → 푃. For any 푔 ∈ 퐺, we denote right multiplication by 푔 as 휎푔 ∶ 푃 → 푃; and for every 푝 ∈ 푃, we denote the orbit map as 휎푝 ∶ 퐺 → 푃. Given a smooth function 푓 ∶ 푀 → 푁 between manifolds, we denote the tangent map at some 푥 ∈ 푀 as 푇푥푓 ∶ 푇푥푀 → 푇푓(푥)푁. This is to explicitly show the functoriality of 푇푥. 1 Connections on principal bundles 1.1 Connections as horizontal distributions Recall that a vector 푣 ∈ 푇푝푃 is called vertical if 푇푝휋(푣) = 0.
    [Show full text]