Natural Operations in Differential Geometry

Capa
Springer 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

PREFACE
1
MANIFOLDS AND LIE GROUPS
4
Submersions and immersions
11
Vector fields and flows
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
References
417
List of symbols
428
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