Natural Operations in Differential Geometry

1. Front Matter

   Pages i-3

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2. Manifolds and Lie Groups

   • Ivan Kolář, Jan Slovák, Peter W. Michor

   Pages 4-48

3. Differential Forms

   • Ivan Kolář, Jan Slovák, Peter W. Michor

   Pages 49-75

4. Bundles and Connections

   • Ivan Kolář, Jan Slovák, Peter W. Michor

   Pages 76-115

5. Jets and Natural Bundles

   • Ivan Kolář, Jan Slovák, Peter W. Michor

   Pages 116-167

6. Finite Order Theorems

   • Ivan Kolář, Jan Slovák, Peter W. Michor

   Pages 168-211

7. Methods for Finding Natural Operators

   • Ivan Kolář, Jan Slovák, Peter W. Michor

   Pages 212-248

8. Further Applications

   • Ivan Kolář, Jan Slovák, Peter W. Michor

   Pages 249-295

9. Product Preserving Functors

   • Ivan Kolář, Jan Slovák, Peter W. Michor

   Pages 296-328

10. Bundle Functors on Manifolds

    • Ivan Kolář, Jan Slovák, Peter W. Michor

    Pages 329-349

11. Prolongation of Vector Fields and Connections

    • Ivan Kolář, Jan Slovák, Peter W. Michor

    Pages 350-375

12. General Theory of Lie Derivatives

    • Ivan Kolář, Jan Slovák, Peter W. Michor

    Pages 376-393

13. Gauge Natural Bundles and Operators

    • Ivan Kolář, Jan Slovák, Peter W. Michor

    Pages 394-416

14. Back Matter

    Pages 417-434

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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.

• Category over Manifolds
• Jet
• Natural Bundle
• Natural Operator
• differential geometry
• manifold
• mathematical physics

• Department of Algebra and Geometry, Faculty of Science, Masaryk University, Brno, Czechoslovakia

  Ivan Kolář, Jan Slovák

• Institut für Mathematik, Universität Wien, Wien, Austria

  Peter W. Michor

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