Introduction to Global Variational Geometry, Volume 6

1st Edition

Authors: Demeter Krupka
Hardcover ISBN: 9780444876034
eBook ISBN: 9780080954158
Imprint: Elsevier Science
Published Date: 1st September 1985
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Table of Contents

Tentative Table of Contents: Preface List of Standard Symbols Chapter 1: Smooth Manifolds

  1. Smooth Manifolds 1.1. Smooth Manifolds 1.2. Smooth Mappings 1.3. Contact of Smooth Mappings 1.4. Jet of a Mapping 1.5. Composition of Mappings 1.6. Submanifolds 1.7. Product of Manifolds
  2. The Tangent Bundle 2.1. Tangent Vectors 2.2. Tangent Bundle 2.3. Tangent Mapping
  3. Tensor Bundles 3.1. Cotangent Bundle 3.2. Tensor Bundles 3.3. Fibered Product of Tensor Bundles 3.4. Induced Morphisms of Tensor Bundles 3.5. Tensor Fields
  4. The Rank of a Mapping 4.1. The Rank Theorem 4.2. Immersions 4.3. Submersions
  5. Fibered Manifolds 5.1. Fibered Manifolds 5.2. Morphisms of Fibered Manifolds 5.3. Vertical Vectors, Horizontal Forms 5.4. Fibrations 5.5. Sections of Fibered Manifolds 5.6. Vector Bundles 5.7. Vector Bundle Morphisms 5.8. Inverse Image of a Vector Bundle Notes and Additional Topics Chapter 2: Analysis on Manifolds
  6. Vector Fields 1.1. Vector Fields 1.2. Local Flow 1.3. Global Flow 1.4. Differential Equations
  7. Differential Forms 2.1. Differential Forms 2.2. Exterior Derivative 2.3. The Poincare Lemma 2.4. Contraction of a Form by a Vector Field 2.5. The Lie Derivative
  8. Distributions 3.1. Vector Distributions 3.2. Distributions Generated by Forms 3.3. Complete Integrability 3.4. Different


This book provides a comprehensive introduction to modern global variational theory on fibred spaces. It is based on differentiation and integration theory of differential forms on smooth manifolds, and on the concepts of global analysis and geometry such as jet prolongations of manifolds, mappings, and Lie groups.

The book will be invaluable for researchers and PhD students in differential geometry, global analysis, differential equations on manifolds, and mathematical physics, and for the readers who wish to undertake further rigorous study in this broad interdisciplinary field.

Featured topics

  • Analysis on manifolds
  • Differential forms on jet spaces
  • Global variational functionals
  • Euler-Lagrange mapping
  • Helmholtz form and the inverse problem
  • Symmetries and the Noether’s theory of conservation laws
  • Regularity and the Hamilton theory
  • Variational sequences
  • Differential invariants and natural variational principles

Key Features

  • First book on the geometric foundations of Lagrange structures
  • New ideas on global variational functionals
  • Complete proofs of all theorems
  • Exact treatment of variational principles in field theory, inc. general relativity
  • Basic structures and tools: global analysis, smooth manifolds, fibred spaces


This book is mainly intended for Universities, research institutions and libraries. It is a basic work on foundations of a mathematical discipline, the first new book on the geometry of the calculus of variations on manifolds, and contains up-to-date research: primary Mathematics Subject Classification 58 (Global Analysis).

It is furthermore intended for individuals (researchers, post-doctorals, PhD students), as the book supports research in geometry, global analysis, mathematical and theoretical physics, field theory, and mechanics and summarizes journal papers on global variational theory, published during the last 40 years.


© Elsevier Science 1985
Elsevier Science
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About the Authors

Demeter Krupka Author

Affiliations and Expertise

Palacky University, Department of Algebra and Geometry, Olomouc, Czech Republic