## Natural Operations in Differential GeometrySpringer Science & Business Media, 9 de mar. de 2013 - 434 páginas The aim of this work is threefold: First it should be a monographical work on natural bundles and natural op erators in differential geometry. This is a field which every differential geometer has met several times, but which is not treated in detail in one place. Let us explain a little, what we mean by naturality. Exterior derivative commutes with the pullback of differential forms. In the background of this statement are the following general concepts. The vector bundle A kT* M is in fact the value of a functor, which associates a bundle over M to each manifold M and a vector bundle homomorphism over f to each local diffeomorphism f between manifolds of the same dimension. This is a simple example of the concept of a natural bundle. The fact that exterior derivative d transforms sections of A kT* M into sections of A k+1T* M for every manifold M can be expressed by saying that d is an operator from A kT* M into A k+1T* M. |

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### Conteúdo

1 | |

4 | |

11 | |

16 | |

Lie groups | 30 |

Lie subgroups and homogeneous spaces | 41 |

CHAPTER II DIFFERENTIAL FORMS | 49 |

Differential forms | 61 |

Polynomial GLVequivariant maps | 213 |

Natural operators on linear connections the exterior differential | 220 |

The tensor evaluation theorem | 223 |

Generalized invariant tensors | 230 |

The orbit reduction | 233 |

The method of differential equations | 245 |

FURTHER APPLICATIONS | 249 |

Two problems on general connections | 255 |

Derivations on the algebra of differential forms and the FrölicherNijenhuis bracket | 67 |

BUNDLES AND CONNECTIONS | 76 |

Principal fiber bundles and Gbundles | 86 |

Principal and induced connections | 99 |

JETS AND NATURAL BUNDLES | 116 |

Jets | 117 |

Jet groups | 128 |

Natural bundles and operators | 138 |

Prolongations of principal fiber bundles | 149 |

Canonical differential forms | 154 |

Connections and the absolute differentiation | 158 |

FINITE ORDER THEOREMS | 168 |

Bundle functors and natural operators | 169 |

Peetrelike theorems | 176 |

The regularity of bundle functors | 185 |

Actions of jet groups | 192 |

The order of bundle functors | 202 |

The order of natural operators | 205 |

METHODS FOR FINDING NATURAL OPERATORS | 212 |

Topics from Riemannian geometry | 265 |

Multilinear natural operators | 280 |

PRODUCT PRESERVING FUNCTORS | 296 |

Product preserving functors | 308 |

Examples and applications | 318 |

BUNDLE FUNCTORS ON MANIFOLDS | 329 |

The flownatural transformation | 336 |

Star bundle functors | 345 |

Prolongations of vector fields to Weil bundles | 351 |

The case of the second order tangent vectors | 357 |

Prolongations of connections to FY M | 363 |

The cases FY FM and FY Y | 369 |

GENERAL THEORY OF LIE DERIVATIVES | 376 |

Lie derivatives of morphisms of fibered manifolds | 387 |

GAUGE NATURAL BUNDLES AND OPERATORS | 394 |

Base extending gauge natural operators | 405 |

417 | |

428 | |

### Outras edições - Visualizar todos

Natural Operations in Differential Geometry Ivan Kolar,Peter W. Michor,Jan Slovak Visualização parcial - 1993 |

Natural Operations in Differential Geometry Ivan Kolar,Peter W. Michor,Jan Slovak Não há visualização disponível - 2010 |

Natural Operations in Differential Geometry Ivan Kolar,Peter W. Michor,Jan Slovak Não há visualização disponível - 2014 |

### Termos e frases comuns

action of G arbitrary associated bundle associated maps bijection bilinear bracket bundle atlas called canonical chart cocycle commutes component consider construction coordinate expression corresponding covariant curvature curve deduce defined definition denote determined diffeomorphism differential dimension equations equivariance fibered manifold finite dimensional finite order formula functor F G-equivariant gauge natural bundle gauge natural operators geometric given GL(m GL(n GL(V Hence homogeneous homotheties implies induced invariant tensor theorem isomorphism jet group Kolár left action Lemma Let F Lie algebra Lie derivative Lie group linear connection linear map map f metric morphism multiplication natural transformations neighborhood order natural operators polynomial principal bundle principal connection principal fiber bundle projectable vector field Proof Proposition Pt(c r-jets r-th order respect restriction Riemannian satisfying smooth function smooth mapping standard fiber structure group subalgebra subgroup submanifold subspace tangent bundle unique vector space zero