.

0 (hodnocen0 x )

BK

1. vyd.

V Brně : Univerzita J.E. Purkyně, 1990

193 s. ; 25 cm

ISBN 80-210-0165-8 (váz.)

Folia facultatis scientiarum naturalium Universitatis Purkynianae Brunensis. Mathematica = Přírodovědecká fakulta Univerzity J.E. Purkyně v Brně ; 1

Obsahuje bibliografii a rejstřík

000053097

List of standard symbols 7 // Preface 9 // Part 1 // ELEMENTARY THEORY OF DIFFERENTIAL INVARIANTS // 1. Lie groups 11 // 1.1. Lie groups 11 // 1.2. Semi-direct products of Lie groups 16 // 1.3. Lie group actions 19 // 2. Differential invariants 36 // 2.1. Manifolds of jets 36 // 2.2. Higher order frames 39 // 2.3. Fundamental categories 41 // 2.4. Differential invariants and their realizations 44 // 2.5. Natural transformations of liftings, associated with the r-frame lifting 46 // 3. Differential invariants and Lie derivatives 47 // 3.1. Jets of sections of a submersion 47 // 3.2. Lie algebras of differential groups 50 // 3.3. Lifting and fundamental vector fields 56 // 3.4. Differential invariants and Lie derivatives 58 // 4. Invariant tensors 66 // 4.1. Absolute invariant tensors 66 // 4.2. Characters of the general linear group 71 // 4.3. Relative invariant tensors 73 // 4.4. Multilinear invariants of the general linear group 80 // 5. Prolongations of liftings 84 // 5.1. Prolongations of Lie groups 84 // [5] // 5.2. Prolongations of left G-manifolds 86 // 5.3. Prolongations of a principal G-bundle 86 // 5.4. Prolongations of a fiber bundle 89 // 5.5. Prolongations of the r-frame lifting and of the associated liftings 92 // 5.6. Natural differential operators 95 // 6. Fundamental vector fields on prolongations of GZ.„(/?)-modules 97 // 6.1. Projectable vector fields and their prolongations 97

// 6.2. Fundamental vectors fields on prolongations of GL„(i?)-moduIes 101 // 6.3. Lie bracket of fundamental vector fields on prolongations of GLn(i?)-modules . . 106 // 7. The structure of differential groups 110 // 7.1. Structure constants of a differential group 110 // 7.2. Vector spaces generating the Lie algebra of a differential group 116 // 7.3. The semi-direct product structure of a differential group and normal subgroups . 118 // 7.4. Differential invariants with values in GZ.n(7?)-manifolds 129 // Part 2 // NATURAL GEOMETRIC OPERATIONS: EXAMPLES // 8. Natural differential operators between tensor bundles 131 // 8.1. Globally defined homogeneous functions 131 // 8.2. Natural differential operators of order zero 134 // 8.3. Natural differential operators of higher orders 141 // 8.4. The uniqueness of exterior derivative 146 // 8.5. Bilinear natural differential operators on vector valued forms 148 // 9. Geometric objects naturally induced from metric 154 // 9.1. The uniqueness of the Levi —Civita connection 154 // 9.2. Natural connections of higher order 157 // 9.3. Natural prolongations of Riemannian metrics on manifolds to metrics on tangent // bundles 160 // 10. Other natural differential operators 166 // 10.1. Natural transformations of the second order tangent functor 167 // 10.2. Natural lifts of vector fields 169 // 10.3. Principal connections on frame bundles 174 // 10.4. Natural operations with linear connections 178 // References 187 // Index 191

(OCoLC)24380399

cnb000058163