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Natural Operations in Differential Geometry

By Kolar, Ivan

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Book Id: WPLBN0000697755
Format Type: PDF eBook
File Size: 2.81 MB
Reproduction Date: 2005

Title: Natural Operations in Differential Geometry  
Author: Kolar, Ivan
Volume:
Language: English
Subject: Science., Mathematics, Logic
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Kolar, I. (n.d.). Natural Operations in Differential Geometry. Retrieved from http://www.hawaiilibrary.net/


Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: The aim of this work is threefold: First it should be a monographical work on natural bundles and natural operators 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 neutrality.

Table of Contents
TABLE OF CONTENTS PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER I. MANIFOLDS AND LIE GROUPS . . . . . . . . . . . . . . . . 4 1. Di erentiable manifolds . . . . . . . . . . . . . . . . . . . . . 4 2. Submersions and immersions . . . . . . . . . . . . . . . . . . 11 3. Vector elds and ows . . . . . . . . . . . . . . . . . . . . . 16 4. Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5. Lie subgroups and homogeneous spaces . . . . . . . . . . . . . 41 CHAPTER II. DIFFERENTIAL FORMS . . . . . . . . . . . . . . . . . . . 49 6. Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . 49 7. Di erential forms . . . . . . . . . . . . . . . . . . . . . . . 61 8. Derivations on the algebra of di erential forms and the Frolicher-Nijenhuis bracket . . . . . . . . . . . . . . . 67 CHAPTER III. BUNDLES AND CONNECTIONS . . . . . . . . . . . . . . . 76 9. General ber bundles and connections . . . . . . . . . . . . . . 76 10. Principal ber bundles and G-bundles . . . . . . . . . . . . . . 86 11. Principal and induced connections . . . . . . . . . . . . . . . 99 CHAPTER IV. JETS AND NATURAL BUNDLES . . . . . . . . . . . . . . . 116 12. Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 13. Jet groups . . . . . . . . . . . . . . . . . . . . . . . . . . 128 14. Natural bundles and operators . . . . . . . . . . . . . . . . . 138 15. Prolongations of principal ber bundles . . . . . . . . . . . . . 149 16. Canonical di erential forms . . . . . . . . . . . . . . . . . . 154 17. Connections and the absolute di erentiation . . . . . . . . . . . 158 CHAPTER V. FINITE ORDER THEOREMS . . . . . . . . . . . . . . . . . 168 18. Bundle functors and natural operators . . . . . . . . . . . . . . 169 19. Peetre-like theorems . . . . . . . . . . . . . . . . . . . . . . 176 20. The regularity of bundle functors . . . . . . . . . . . . . . . . 185 21. Actions of jet groups . . . . . . . . . . . . . . . . . . . . . . 192 22. The order of bundle functors . . . . . . . . . . . . . . . . . . 202 23. The order of natural operators . . . . . . . . . . . . . . . . . 205 CHAPTER VI. METHODS FOR FINDING NATURAL OPERATORS . . . . . . 212 24. Polynomial GL(V )-equivariant maps . . . . . . . . . . . . . . 213 25. Natural operators on linear connections, the exterior di erential . . 220 26. The tensor evaluation theorem . . . . . . . . . . . . . . . . . 223 27. Generalized invariant tensors . . . . . . . . . . . . . . . . . . 230 28. The orbit reduction . . . . . . . . . . . . . . . . . . . . . . 233 29. The method of di erential equations . . . . . . . . . . . . . . 245

 

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