Global Theory of Connections and Holonomy Groups

ISBN-13: 9789401015523 / Angielski / Miękka / 2011 / 264 str.

Andre Lichnerowicz

Opis

This work was conceived as an introduction to global differ- ential geometry. It assumes known only the elements of classical differential geometry and Lie groupst. Some theorems are admit- ted without proof, but in the majority of cases detailed proofs are given. If this book allows researchers to initiate them- selves in contemporary works on the global theory of connections, it will have achieved its goal. The Consiglio Nazionale delle Ricerche has done me the great honour of including my book in its fine collection. I would wish it to find here an expression of my profound gratitude. Monsieur Dalla Volta has graciously provided a skilful and invaluable cooperation with the material cares of publication, which has been a great help to me. Without a doubt this book would never have seen the light of day without the illuminating advice of Monsieur Enrico Bompiani; it was conceived during the course of some weeks spent in 1955 at the University of Rome in the unforgettable atmosphere of the Istituto di Matematica. A. LICHNEROWICZ t The notations used for linear groups are those of Chevalley (Lie Groups) .

Spis treści

I General Notions about Differentiable Manifolds.- 1•1: Differentiable Manifold.- 1•1.1 Atlas: Differentiable Manifold.- .2 Scalar Function Defined on a Manifold.- .3 Mapping of Class C?.- .4 Tangent Vector Spaces at a Point.- . 5 Frames and Coframes.- .6 Pseudoscalars, Orientations, Tensors in a Differentiable Manifold.- .7 The Notion of Differentiable Fibre Bundle.- .8 The Fibre Bundles Attached to a Differentiable Manifold.- .9 Riemannian Manifolds: Whitney’s Theorem.- . 10 Images under a Mapping.- 1•2: Exterior Differential Forms.- 1•2.11 The Kronecker Tensor.- .12 The Space of q-Forms ?x (q).- .13 The Exterior Product.- .14 Exterior Product of Linear q-Forms: Expression for and Value of a q-Form.- .15 Two Results on Exterior Products of Linear Forms.- .16 Reduction of an Exterior Quadratic Form.- .17 Exterior Differential Forms.- .18 Exterior Differential of a q-Form.- .19 Inverse Image of a Form under a Mapping.- 1•2.20 Closed Forms — Local Study.- .21 Pfaffian Systems: Frobenius’ Theorem.- 1•3: Vector Valued Forms.- 1•3.22 Notion of a Vector Valued Form.- .23 Exterior Differential of a Form ?.- .24 Case Where the Vector Space Admits a Lie-Algebraic Structure.- II Infinitesimal Connections: Linear Connections.- 2•1: Homotopy Notions.- 2•1.25 Paths: Homotopy: Poincaré Group.- .26 Factorisation Lemma.- 2•2: Infinitesimal Connections on a Principal Fibre Bundle.- 2•2.27 Principal Fibre Bundle and the Lie Algebra of the Structural Group.- .28 First Definition of an Infinitesimal Connection.- .29 Second Definition of an Infinitesimal Connection.- . 30 Development.- . 31 Local Sections.- .32 Holonomy Groups of an Infinitesimal Connection..- .33 Tensors and Tensor Forms on E.- .34 Passage From One Connection to Another.- .35 Absolute Differential of a q-Form: Curvature of an Infinitesimal Connection.- .36 Expression for the Curvature Form.- 2•3: Linear Connections.- 2•3.37 Notion of Linear Connection.- .38 Explicit Formulae.- .39 The Absolute Differential in a Linear Connection.- .40 Torsion of a Linear Connection.- .41 Curvature of a Linear Connection.- .42 The Bianchi Identities for a Linear Connection.- 2•3.43 Explicit Formulae in Arbitrary Frames and in Local Coordinates.- .44 The Ricci Identity.- .45 Fibre Bundle of Affine Frames.- .46 Affine Connection Associated with a Linear Connection.- .47 Transport Relative to a Linear Connection: Holonomy Groups.- .48 Image of a Linear Connection.- .49 Development on Affine Spaces.- .50 Geodesics.- 2•4: Riemannian Connections.- 2•4.51 Notion of Euclidian Connection: Holonomy Group.- .52 Riemannian Connections.- .53 Properties of the Riemannian Connection.- III Holonomy Groups and Curvature.- 3•1: General Case and Manifolds with a Linear Connection.- 3•1.54 Transport of a Tensor: Tensor with Vanishing Covariant Derivative.- .55 Local Holonomy Group.- .56 Special Local Sections.- .57 Elements of the Lie Algebra of 1.- .58 elements of the lie algebra of 2.- .59 Case of an Infinitesimal Connection on a Principal Fibre Bundle.- .60 Holonomy Group and Curvature.- .61 Case of a Linear Connection.- .62 Notion of Infinitesimal Holonomy Group.- .63 Singular Points for Infinitesimal Holonomy.- .64 Regular Points for Infinitesimal Holonomy.- .65 Points Singular for Local Holonomy.- .66 Connected Components of Holonomy of Vn.- .67 Real Analytic Manifold with an Analytic Connection.- .68 Study of the Group ?x(Vn) in the Case Where ?x(Vn) is Irreducible.- 3•2: Riemannian Manifolds: Reducibility.- 3•2.69 Holonomy Groups.- .70 Reducibility of a Riemannian Manifold.- .71 Complete Reducibility Of ?x.- . 72 Study of the Group ?x.- .73 Study of the Group ?x.- . 74 Geodesic Normal Coordinates.- .75 Complete Riemannian Manifolds: A Theorem by Georges De Rham.- .76 The Group ?x for a Complete Riemannian Manifold.- IV Harmonic Forms and Forms with Zero Covariant Derivative.- 4·1: Elements of Homology.- 4•1.77 Differentiable Chains.- . 78 Boundary.- .79 Integral of a Form: Stokes’ Formula.- .80 Homology on Forms.- 4•2: Harmonic Forms.- 4•2.81 The Volume Element Form on Vn.- .82 The Hodge *-Operator on p-Forms.- .83 The Operators ? and ?.- .84 The Global Scalar Product on a Compact Manifold and Harmonic Forms.- .85 Fundamental Theorems on Harmonic Forms.- 4•3: The Operators Defined by a Form on a Riemannian Manifold.- 4•3.86 Definition of the Operators Kh.- .87 Relations between Kh and Kk-h.- .88 Case Where F has Zero Covariant Derivative: Relations With d and ?.- .89 The Operators Kh and the Operator ?.- .90 Example: Case Where the Degree of F is Twice an Odd Number.- V Almost Complex Manifolds and Subordinate Structures.- 5•1: Complex Structure on a Real Vector Space.- 5•1.91 Complexification of a Real Vector Space.- .92 Complex Structure on a Real Space.- .93 Bases of T2n Adapted to a Complex Structure of T2n.- .94 The Operators C and M on Forms.- 5•2: Hermitian Vector Spaces.- 5•2.95 Notion of Hermitian Vector Space.- .96 Hermitian Structure Subordinate to an Exterior Quadratic Form.- .97 Bases Adapted to a Hermitian Structure.- .98 The Operators L and ? for a Hermitian Vector Space.- .99 Hodge-Lepage Decomposition for a q-Form.- 5•3: Almost Complex Structure and Subordinate Structures on a Differentiable Manifold.- 5•3.100 Manifold with Complex Analytic Structure.- .101 Manifold with Almost Complex Structure.- .102 Torsion of an Almost Complex Structure.- .103 Integrability of an Almost Complex Structure.- .104 Calculation of the Torsion Tensor of an Almost Complex Structure.- .105 Almost Complex Torsion and Vector Fields.- . 106 Almost Hermitian Structure.- 5•4: Almost Complex Connections.- 5•4.107 Complex Linear Connections.- .108 Notion of an Almost Complex Connection.- .109 Real Linear and Almost Complex Connections.- .110 Almost Complex Torsion and Connections.- .111 Almost Hermitian Connections.- .112 Second Canonical Connection of an Almost Hermitian Manifold.- .113 Case of Hermitian and Pseudohermitian Manifolds.- .114 Case of Pseudokählerian Manifolds.- 5•4.115 Quadratic Form with Vanishing Covariant Derivative on A Riemannian Manifold.- .116 Kählerian Manifolds.- .117 Examples of Kählerian Manifolds.- .118 Properties Concerning Holonomy Groups.- .119 Reducibility of Pseudokählerian Manifolds.- 5•5: Forms on Pseudohermitian and Pseudokählerian Manifolds.- 5•5.120 Orthogonality on an Almost Hermitian Manifold.- .121 The Operators d? and d? on an Almost Complex Manifold.- .122 The Operators ?? and ?? on a pseudohermitian Manifold.- .123 On the Operators M, d and ? on a Pseudohermitian Manifold.- .124 Operators on a Pseudokählerian Manifold.- .125 Global Properties of Compact Pseudokählerian Manifolds.- List of Symbols.