DOCSLIB.ORG

Advanced Differential Geometry for Theoreticians. Fiber Bundles, Jet

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Advanced Differential Geometry for Theoreticians. Fiber Bundles, Jet ! " # $ ! " # %# & ' () *! # ! # ' +,-.& !%!(+,/0*%!! (+,,/*& # %# & ' # .00 " 1.2 " # ! !"!" #$ %#$ ! " # $ !% ! & $ ' ' ($ ' # % % ) %* %' $ ' + " % & ' ! # $, ( $ - . ! "- ( % . % % % % $ $ $ - - - - // $$$ 0 1"1"#23." "0" )*4/ +) * !5 !& 6!7%66898& % ) - 2 : ! * & #&'()*+,+-.(+,)*/.+) /- ;9<8="0" )*4/ +) "3 " & 9<8= Contents Introduction 5 1 Geometry of fibre bundles 7 1.1 Fibre bundles ........................ 7 1.2 Vector and affine bundles .................. 15 1.3Vectorfields......................... 22 1.4Exteriorandtangent-valuedforms............. 25 2 Jet manifolds 33 2.1Firstorderjetmanifolds.................. 33 2.2Higherorderjetmanifolds................. 35 2.3Differentialoperatorsandequations............ 42 2.4Infiniteorderjetformalism................. 45 3 Connections on fibre bundles 51 3.1Connectionsastangent-valuedforms........... 51 3.2 Connections as jet bundle sections ............. 54 3.3Curvatureandtorsion................... 57 3.4Linearandaffineconnections............... 58 3.5Flatconnections....................... 63 3.6 Connections on composite bundles ............. 64 4 Geometry of principal bundles 69 4.1GeometryofLiegroups................... 69 4.2Bundleswithstructuregroups............... 73 1 2 CONTENTS 4.3 Principal bundles ...................... 76 4.4Principalconnections.................... 81 4.5Canonicalprincipalconnection............... 86 4.6Gaugetransformations................... 88 4.7 Geometry of associated bundles .............. 91 4.8Reducedstructure...................... 95 5 Geometry of natural bundles 99 5.1 Natural bundles ....................... 99 5.2Linearworldconnections.................. 103 5.3Affineworldconnections.................. 108 6 Geometry of graded manifolds 113 6.1Grassmann-gradedalgebraiccalculus........... 113 6.2Grassmann-gradeddifferentialcalculus.......... 117 6.3Gradedmanifolds...................... 122 6.4Gradeddifferentialforms.................. 128 7 Lagrangian theory 131 7.1Variationalbicomplex.................... 131 7.2 Lagrangian theory on fibre bundles ............ 133 7.3Grassmann-gradedLagrangiantheory........... 142 7.4Noetheridentities...................... 152 7.5Gaugesymmetries..................... 159 8 Topics on commutative geometry 163 8.1Commutativealgebra.................... 163 8.2Differentialoperatorsonmodules............. 169 8.3Homologyandcohomologyofcomplexes......... 172 8.4Differentialcalculusoveracommutativering....... 175 8.5Sheafcohomology...................... 179 8.6Local-ringedspaces..................... 188 3 Bibliography 195 Index 198 4 Introduction In contrast with quantum field theory, classical field theory can be formu- lated in a strict mathematical way by treating classical fields as sections of smooth fibre bundles [9, 17, 21, 24]. This also is the case of time- dependent non-relativistic mechanics on fibre bundles over R [10, 16, 19]. This book aim to compile the relevant material on fibre bundles, jet manifolds, connections, graded manifolds and Lagrangian theory [9, 17, 22]. Thebookisbasedonthegraduateandpostgraduatecoursesoflec- tures given at the Department of Theoretical Physics of Moscow State University (Russia). It addresses to a wide audience of mathematicians, mathematical physicists and theoreticians. It is tacitly assumed that the reader has some familiarity with the basics of differential geometry [11, 13, 26]. Throughout the book, all morphisms are smooth (i.e. of class C∞) and manifolds are smooth real and finite-dimensional. A smooth real manifold is customarily assumed to be Hausdorff and second-countable (i.e., it has a countable base for topology). Consequently, it is a lo- cally compact space which is a union of a countable number of compact subsets, a separable space (i.e., it has a countable dense subset), a para- compact and completely regular space. Being paracompact, a smooth manifold admits a partition of unity by smooth real functions. Unless otherwise stated, manifolds are assumed to be connected (and, conse- quently, arcwise connected). We follow the notion of a manifold without boundary. 5 6 Chapter 1 Geometry of fibre bundles Throughout the book, fibre bundles are smooth finite-dimensional and locally-trivial. 1.1 Fibre bundles Let Z be a manifold. By → ∗ ∗ → πZ : TZ Z, πZ : T Z Z are denoted its tangent and cotangent bundles, respectively. Given coordinates (zα)onZ, they are equipped with the holonomic coordinates ∂zλ (zλ, z˙λ), z˙λ = z˙μ, ∂zμ ∂zμ (zλ, z˙ ), z˙ = z˙ , λ λ ∂zλ μ λ with respect to the holonomic frames {∂λ} and coframes {dz } in the tangent and cotangent spaces to Z, respectively. Any manifold mor- phism f : Z → Z yields the tangent morphism ∂fλ Tf : TZ → TZ, z˙λ ◦ Tf = z˙μ. ∂xμ Let us consider manifold morphisms of maximal rank. They are im- mersions (in particular, imbeddings) and submersions. An injective im- mersion is a submanifold, and a surjective submersion is a fibred manifold (in particular, a fibre bundle). 7 8 CHAPTER 1. GEOMETRY OF FIBRE BUNDLES Given manifolds M and N,bytherank of a morphism f : M → N at a point p ∈ M is meant the rank of the linear morphism Tpf : TpM → Tf(p)N. For instance, if f is of maximal rank at p ∈ M,thenTpf is injective when dim M ≤ dim N and surjective when dim N ≤ dim M. In this case, f is called an immersion and a submersion at a point p ∈ M,respectively. Since p → rankpf is a lower semicontinuous function, then the mor- phism Tpf is of maximal rank on an open neighbourhood of p,too.It follows from the inverse function theorem that: • if f is an immersion at p, then it is locally injective around p. • if f is a submersion at p, it is locally surjective around p. If f is both an immersion and a submersion, it is called a local diffeo- morphism at p. In this case, there exists an open neighbourhood U of p such that f : U → f(U) is a diffeomorphism onto an open set f(U) ⊂ N. A manifold morphism f is called the immersion (resp. submersion) if it is an immersion (resp. submersion) at all points of M. A submersion is necessarily an open map, i.e., it sends open subsets of M onto open subsets of N. If an immersion f is open (i.e., f is a homeomorphism onto f(M) equipped with the relative topology from N), it is called the imbedding. Apair(M,f) is called a submanifold of N if f is an injective im- mersion. A submanifold (M,f)isanimbedded submanifold if f is an imbedding. For the sake of simplicity, we usually identify (M,f)with f(M). If M ⊂ N, its natural injection is denoted by iM : M → N. There are the following criteria for a submanifold to be imbedded. Theorem 1.1.1:Let(M,f) be a submanifold of N. (i) The map f is an imbedding iff, for each point p ∈ M, there exists a (cubic) coordinate chart (V,ψ)ofN centered at f(p) so that f(M)∩V consists of all points of V with coordinates (x1,...,xm, 0,...,0). 1.1. FIBRE BUNDLES 9 (ii) Suppose that f : M → N is a proper map, i.e., the pre-images of compact sets are compact. Then (M,f) is a closed imbedded submani- fold of N. In particular, this occurs if M is a compact manifold. (iii) If dim M = dim N,then(M,f) is an open imbedded submanifold of N. 2 A triple π : Y → X, dim X = n>0, (1.1.1) is called a fibred manifold if a manifold morphism π is a surjective sub- mersion, i.e., the tangent morphism Tπ : TY → TX is a surjection. One says that Y is a total space of a fibred manifold (1.1.1), X is its base, π −1 is a fibration,andYx = π (x)isafibre over x ∈ X. Any fibre is an imbedded submanifold of Y of dimension dim Y − dim X. Theorem 1.1.2: A surjection (1.1.1) is a fired manifold iff a manifold λ i λ Y admits an atlas of coordinate charts (UY ; x ,y ) such that (x )are coordinates on π(UY ) ⊂ X and coordinate transition functions read xλ = f λ(xμ),yi = f i(xμ,yj). These coordinates are called fibred coordinates compatible with a fibra- tion π. 2 By a local section of a surjection (1.1.1) is meant an injection s : U → Y of an open subset U ⊂ X such that π ◦ s =IdU, i.e., a section sends any point x ∈ X into the fibre Yx over this point. A local section also is defined over any subset N ∈ X as the restriction to N of a local section over an open set containing N.IfU = X, one calls s the global section. Hereafter, by a section is meant both a global section and a local section (overanopensubset). Theorem 1.1.3: A surjection π (1.1.1) is a fibred manifold iff, for each point y ∈ Y , there exists a local section s of π : Y → X passing through y. 2 10 CHAPTER 1. GEOMETRY OF FIBRE BUNDLES The range s(U)ofalocalsections : U → Y of a fibred manifold Y → X is an imbedded submanifold of Y .Italsoisaclosed map,which sends closed subsets of U onto closed subsets of Y .Ifs is a global section, then s(X) is a closed imbedded submanifold of Y .Globalsectionsofa fibred manifold need not exist. Theorem 1.1.4:LetY → X be a fibred manifold whose fibres are diffeomorphic to Rm. Any its section over a closed imbedded submanifold (e.g., a point) of X is extended to a global section. In particular, such a fibred manifold always has a global section. 2 λ i Given fibred coordinates (UY ; x ,y ), a section s of a fibred manifold Y → X is represented
Load more

Recommended publications
  • Jet Fitting 3: a Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces Via Polynomial Fitting Frédéric Cazals, Marc Pouget
    Jet fitting 3: A Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces via Polynomial Fitting Frédéric Cazals, Marc Pouget To cite this version: Frédéric Cazals, Marc Pouget. Jet fitting 3: A Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces via Polynomial Fitting. ACM Transactions on Mathematical Software, Association for Computing Machinery, 2008, 35 (3). inria-00329731 HAL Id: inria-00329731 https://hal.inria.fr/inria-00329731 Submitted on 2 Mar 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Jet fitting 3: A Generic C++ Package for Estimating the Differential Properties on Sampled Surfaces via Polynomial Fitting FRED´ ERIC´ CAZALS INRIA Sophia-Antipolis, France. and MARC POUGET INRIA Nancy Gand Est - LORIA, France. 3 Surfaces of R are ubiquitous in science and engineering, and estimating the local differential properties of a surface discretized as a point cloud or a triangle mesh is a central building block in Computer Graphics, Computer Aided Design, Computational Geometry, Computer Vision. One strategy to perform such an estimation consists of resorting to polynomial fitting, either interpo- lation or approximation, but this route is difficult for several reasons: choice of the coordinate system, numerical handling of the fitting problem, extraction of the differential properties.
    [Show full text]
  • 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.
    [Show full text]
  • On Galilean Connections and the First Jet Bundle
    ON GALILEAN CONNECTIONS AND THE FIRST JET BUNDLE JAMES D.E. GRANT AND BRADLEY C. LACKEY Abstract. We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordinary differential equations under time-preserving transformations { sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces \laboratory" coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse { the \fundamental theorem" { that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion. 1. Introduction The geometry of a system of ordinary differential equations has had a distinguished history, dating back even to Lie [10]. Historically, there have been three main branches of this theory, depending on the class of allowable transformations considered. The most studied has been differ- ential equations under contact transformation; see x7.1 of Doubrov, Komrakov, and Morimoto [5], for the construction of Cartan connections under this class of transformation. Another classical study has been differential equations under point-transformations. (See, for instance, Tresse [13]). By B¨acklund's Theorem, this is novel only for a single second-order dif- ferential equation in one independent variable. The construction of the Cartan connection for this form of geometry was due to Cartan himself [2]. See, for instance, x2 of Kamran, Lamb and Shadwick [6], for a modern \equivalence method" treatment of this case.
    [Show full text]
  • Arxiv:1410.4811V3 [Math.AG] 5 Oct 2016 Sm X N N+K Linear Subspace of P of Dimension Dk, Where N 6 Dk 6 N , See Definition 2.4
    A NOTE ON HIGHER ORDER GAUSS MAPS SANDRA DI ROCCO, KELLY JABBUSCH, AND ANDERS LUNDMAN ABSTRACT. We study Gauss maps of order k, associated to a projective variety X embedded in projective space via a line bundle L: We show that if X is a smooth, complete complex variety and L is a k-jet spanned line bundle on X, with k > 1; then the Gauss map of order k has finite fibers, unless n X = P is embedded by the Veronese embedding of order k. In the case where X is a toric variety, we give a combinatorial description of the Gauss maps of order k, its image and the general fibers. 1. INTRODUCTION N Let X ⊂ P be an n-dimensional irreducible, nondegenerate projective variety defined over an algebraically closed field | of characteristic 0. The (classical) Gauss map is the rational morphism g : X 99K Gr(n;N) that assigns to a smooth point x the projective tangent space of X at ∼ n N x, g(x) = TX;x = P : It is known that the general fiber of g is a linear subspace of P , and that the N morphism is finite and birational if X is smooth unless X is all of P ,[Z93, KP91, GH79]. In [Z93], Zak defines a generalization of the above definition as follows. For n 6 m 6 N − 1, N let Gr(m;N) be the Grassmanian variety of m-dimensional linear subspaces in P , and define Pm = f(x;a) 2 Xsm × Gr(m;N)jTX;x ⊆ La g; where La is the linear subspace corresponding to a 2 Gr(m;N) and the bar denotes the Zariski closure in X × Gr(m;N).
    [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]
  • A Geometric Theory of Higher-Order Automatic Differentiation
    A Geometric Theory of Higher-Order Automatic Differentiation Michael Betancourt Abstract. First-order automatic differentiation is a ubiquitous tool across statistics, machine learning, and computer science. Higher-order imple- mentations of automatic differentiation, however, have yet to realize the same utility. In this paper I derive a comprehensive, differential geo- metric treatment of automatic differentiation that naturally identifies the higher-order differential operators amenable to automatic differ- entiation as well as explicit procedures that provide a scaffolding for high-performance implementations. arXiv:1812.11592v1 [stat.CO] 30 Dec 2018 Michael Betancourt is the principle research scientist at Symplectomorphic, LLC. (e-mail: [email protected]). 1 2 BETANCOURT CONTENTS 1 First-Order Automatic Differentiation . ....... 3 1.1 CompositeFunctionsandtheChainRule . .... 3 1.2 Automatic Differentiation . 4 1.3 The Geometric Perspective of Automatic Differentiation . .......... 7 2 JetSetting ....................................... 8 2.1 JetsandSomeofTheirProperties . 8 2.2 VelocitySpaces .................................. 10 2.3 CovelocitySpaces................................ 13 3 Higher-Order Automatic Differentiation . ....... 15 3.1 PushingForwardsVelocities . .... 15 3.2 Pulling Back Covelocities . 20 3.3 Interleaving Pushforwards and Pullbacks . ........ 24 4 Practical Implementations of Automatic Differentiation . ............ 31 4.1 BuildingCompositeFunctions. .... 31 4.2 GeneralImplementations . 34 4.3 LocalImplementations. 44
    [Show full text]
  • NATURAL STRUCTURES in DIFFERENTIAL GEOMETRY Lucas
    NATURAL STRUCTURES IN DIFFERENTIAL GEOMETRY Lucas Mason-Brown A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in mathematics Trinity College Dublin March 2015 Declaration I declare that the thesis contained herein is entirely my own work, ex- cept where explicitly stated otherwise, and has not been submitted for a degree at this or any other university. I grant Trinity College Dublin per- mission to lend or copy this thesis upon request, with the understanding that this permission covers only single copies made for study purposes, subject to normal conditions of acknowledgement. Contents 1 Summary 3 2 Acknowledgment 6 I Natural Bundles and Operators 6 3 Jets 6 4 Natural Bundles 9 5 Natural Operators 11 6 Invariant-Theoretic Reduction 16 7 Classical Results 24 8 Natural Operators on Differential Forms 32 II Natural Operators on Alternating Multivector Fields 38 9 Twisted Algebras in VecZ 40 10 Ordered Multigraphs 48 11 Ordered Multigraphs and Natural Operators 52 12 Gerstenhaber Algebras 57 13 Pre-Gerstenhaber Algebras 58 irred 14 A Lie-admissable Structure on Graacyc 63 irred 15 A Co-algebra Structure on Graacyc 67 16 Loday's Rigidity Theorem 75 17 A Useful Consequence of K¨unneth'sTheorem 83 18 Chevalley-Eilenberg Cohomology 87 1 Summary Many of the most important constructions in differential geometry are functorial. Take, for example, the tangent bundle. The tangent bundle can be profitably viewed as a functor T : Manm ! Fib from the category of m-dimensional manifolds (with local diffeomorphisms) to 3 the category of fiber bundles (with locally invertible bundle maps).
    [Show full text]
  • JET TRANSPORT PROPAGATION of UNCERTAINTIES for ORBITS AROUND the EARTH 1 Introduction
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UPCommons. Portal del coneixement obert de la UPC 64rd International Astronautical Congress, Beijing, China. Copyright 2012 by the International Astronautical Federation. All rights reserved. IAC-13. C1.8.10 JET TRANSPORT PROPAGATION OF UNCERTAINTIES FOR ORBITS AROUND THE EARTH Daniel Perez´ ∗, Josep J. Masdemontz, Gerard Gomez´ x Abstract In this paper we present a tool to study the non-linear propagation of uncertainties for orbits around the Earth. The tool we introduce is known as Jet Transport and allows to propagate full neighborhoods of initial states instead of a single initial state by means of usual numerical integrators. The description of the transported neighborhood is ob- tained in a semi-analytical way by means of polynomials in 6 variables. These variables correspond to displacements in the phase space from the reference point selected in an orbit as initial condition. The basis of the procedure is a standard numerical integrator of ordinary differential equations (such as a Runge-Kutta or a Taylor method) where the usual arithmetic is replaced by a polynomial arithmetic. In this way, the solution of the variational equations is also obtained up to high order. This methodology is applied to the propagation of satellite trajectories and to the computation of images of uncertainty ellipsoids including high order nonlinear descriptions. The procedure can be specially adapted to the determination of collision probabilities with catalogued space debris or for the end of life analysis of spacecraft in Medium Earth Orbits. 1 Introduction of a particle that at time t0 is at x0.
    [Show full text]
  • Jets and Differential Linear Logic
    Jets and differential linear logic James Wallbridge Abstract We prove that the category of vector bundles over a fixed smooth manifold and its corresponding category of convenient modules are models for intuitionistic differential linear logic. The exponential modality is modelled by composing the jet comonad, whose Kleisli category has linear differential operators as morphisms, with the more familiar distributional comonad, whose Kleisli category has smooth maps as morphisms. Combining the two comonads gives a new interpretation of the semantics of differential linear logic where the Kleisli morphisms are smooth local functionals, or equivalently, smooth partial differential operators, and the codereliction map induces the functional derivative. This points towards a logic and hence computational theory of non-linear partial differential equations and their solutions based on variational calculus. Contents 1 Introduction 2 2 Models for intuitionistic differential linear logic 5 3 Vector bundles and the jet comonad 10 4 Convenient sheaves and the distributional comonad 17 5 Comonad composition and non-linearity 22 6 Conclusion 27 arXiv:1811.06235v2 [cs.LO] 28 Dec 2019 1 1 Introduction In this paper we study differential linear logic (Ehrhard, 2018) through the lens of the category of vector bundles over a smooth manifold. We prove that a number of categories arising from the category of vector bundles are models for intuitionistic differential linear logic. This is part of a larger project aimed at understanding the interaction between differentiable programming, based on differential λ-calculus (Ehrhard and Regnier, 2003), and differential linear logic with a view towards extending these concepts to a language of non-linear partial differential equations.
    [Show full text]
  • On Galilean Connections and the First Jet Bundle
    Cent. Eur. J. Math. • 10(5) • 2012 • 1889-1895 DOI: 10.2478/s11533-012-0089-4 Central European Journal of Mathematics On Galilean connections and the first jet bundle Research Article James D.E. Grant1∗, Bradley C. Lackey2† 1 Gravitationsphysik, Fakultät für Physik, Universität Wien, Boltzmanngasse 5, 1090 Wien, Austria 2 Trusted Systems Research Group, National Security Agency, 9800 Savage Road, Suite 6511, Fort G.G. Meade, MD 20755, USA Received 30 November 2011; accepted 8 June 2012 Abstract: We see how the first jet bundle of curves into affine space can be realized as a homogeneous space of the Galilean group. Cartan connections with this model are precisely the geometric structure of second-order ordi- nary differential equations under time-preserving transformations – sometimes called KCC-theory. With certain regularity conditions, we show that any such Cartan connection induces “laboratory” coordinate systems, and the geodesic equations in this coordinates form a system of second-order ordinary differential equations. We then show the converse – the “fundamental theorem” – that given such a coordinate system, and a system of second order ordinary differential equations, there exists regular Cartan connections yielding these, and such connections are completely determined by their torsion. MSC: 53C15, 58A20, 70G45 Keywords: Galilean group • Cartan connections • Jet bundles • 2nd order ODE © Versita Sp. z o.o. 1. Introduction The geometry of a system of ordinary differential equations has had a distinguished history, dating back even to Lie [10]. Historically, there have been three main branches of this theory, depending on the class of allowable transformations considered. The most studied has been differential equations under contact transformation; see § 7.1 of Doubrov, Komrakov, and Morimoto [5], for the construction of Cartan connections under this class of transformation.
    [Show full text]
  • Holonomicity in Synthetic Differential Geometry of Jet Bundles
    Holonomicity in Synthetic Differential Geometry of Jet Bundles Hirokazu Nishimura ＩｎｓＭｕtｅ ｏｆ Ｍａtｈｅｍａｈｃｓ、 Ｕｎｉ'ｕｅｒｓｘt!Jof Tｓｕkｕｂａ Ｔｓｕkｕba、IbaTヽａｋｉ、Ｊａｐａｎ Abstract. In the repet山ve appioach to the theory of jet bundles there aie three methods of repetition, which yield non-holonomic, semi-holonomic, and holonomic jet bundles respectively However the classical approach to ｈｏｌｏｎｏｍｌ(ヽJ(ヽtbundles failed to be truly repetitive, 柘r It must resort to such ａ non-repetitive coordinate- dependent construction as Taylor expansion The principal objective ｍ this paper IS to give ａpurely repet山ve definition of holonomicity by using microsquarcs (doｕ- blc tangents), which spells the flatness of the Cartan connection on holoiiomic lnfi- nite jet bundles without effort The defiii由on applies not only to formal manifolds but to microlinear spaces in general, enhancing the applicability of the theory of jet bundles considerably The definition IS shown to degenerate into the classical one ｍ caseヽofformal manifolds Introduction The flatness of the Cartan connection lies smack dab at the center of the theory of infinite jet bundles and its comrade called the geometric theory of nonlinear partial differentialequa- tions Indeed itIS the flatness of the Cartan connection that enables the de Rham complex to decompose two-dimensionally into the variational bicomplcx, from which the algebraic topol- ogy of nonlinear partial differential equations has emerged by means of spectral sequences (ｃｆBocharov et al[1D， just as the algebraic topology of smooth manifolds IS centered
    [Show full text]
  • Math 601 - Vector Bundles Homework II Due March 31
    Math 601 - Vector Bundles Homework II Due March 31 Problem 1 (Dual Bundles) In this problem, F will denote either the field R of real number, or the field C of complex numbers. Given a vector bundle E ! B with fiber Fn, the dual bundle ∗ E = HomF (E; F) is defined using the construction in MS Chapter 3. n a) Say φi : Ui × F ! EjUi are trivializations with U1 \ U2 6= ;. Describe the ∗ −1 transition function for E in terms of the transition function φ2 φ1. Remark 0.1. You'll need to express the dual space as a covariant functor from vec- tor spaces over F (and isomorphisms) to vector spaces over F (and isomorphisms), so that the construction in MS Chapter 3 applies. Also, it's important to notice that in MS the trivializations of E∗ come from (Fn)∗, rather than from F n, so you'll need to compose with the natural isomorphism between Fn and (Fn)∗. b) Consider the natural embedding FP 1 ! FP 2 given by sending [x; y] 2 FP 1 = (F 2 − f0g)=F × to [x; y; 0] 2 FP 2 = (F 3 − f0g)=F ×. There is a natural projection 2 1 π : FP − f[0; 0; 1]g −! FP given by [x; y; z] 7! [x; y; 0] (where we identify FP 1 with its image under the em- bedding), and each fiber of this projection is just a copy of F . Show that this projection is locally trivial over the open sets f[x; y; 0] 2 FP 1 : x 6= 0g and f[x; y; 0] 2 FP 1 : y 6= 0g (hence π is a vector bundle), and compute the tran- sition function defined by the two trivializations.
    [Show full text]