Lectures on differential invariants

CONTENTS   5
List of standard symbols ............................. 7 
Preface ..................................... 9 
Part 1 - ELEMENTARY THEORY OF DIFFERENTIAL INVARIANTS  11
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 
5.2. 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 GLn(R)-modules ........ 97 
6.1. Projectable vector fields and their prolongations ........... 97 
6.2. Fundamental vectors fields on prolongations of GLn(R)-modules ....... 101 
6.3. Lie bracket of fundamental vector fields on prolongations of GLn(R)-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 GLn(R)-manifolds ........... 129 
Part 2 - NATURAL GEOMETRIC OPERATIONS: EXAMPLES   131
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