! " # $ ! " # %# & ' () *! # ! # ' +,-.& !%!(+,/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