# Introduction to Global Variational Geometry, Volume 22

## 1st Edition

**Author:**Demeter Krupka

**eBook ISBN:**9780080954288

**Imprint:**Elsevier Science

**Published Date:**1st January 1980

**Page Count:**500

**View all volumes in this series:**North-Holland Mathematical Library

## 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

1. Vector Fields

1.1. Vector Fields

1.2. Local Flow

1.3. Global Flow

1.4. Differential Equations

2. 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

3. Distributions

3.1. Vector Distributions

3.2. Distributions Generated by Forms

3.3. Complete Integrability

3.4. Differential Ideals

4. Integration of Forms

4.1. Orientation of a Vector Space

4.2. Orientability of Manifolds, Volume Elements

4.3. Piece of a Manifold

4.4. Boundary of a Piece, Induced Volume Elements

4.5. Integration of Forms

4.6. Transformation of an Integral

4.7. Differentiation of an Integral with Respect to a Parameter

4.8. Integration on the Product of Manifolds

4.9. Integration of Exact Forms

5. Integration of Odd Forms

5.1. Odd Scalars, Odd Forms, Volume Elements

5.2. Integration of Odd Forms

5.3. Integration of Functions

5.4. Transformation of an Integral

5.5. Differentiation of an Integral with Respect to a Parameter

5.6. Integration on the Product of Manifolds

5.7. Restriction of a Volume Element to a Submanifold

5.8. Integration of Exact Forms

Notes and Additional Topics

Chapter 3: Lie Transformation Groups

1. Lie Groups

1.1. Lie Groups

1.2. Lie Group Morphisms

1.3. Lie Subgroups

1.4. Normal Subgroups and Quotient Groups

1.5. Invariant Vector Fields

1.6. Lie Algebra of a Lie Group

2. Products of Lie Groups

2.1. Product of Lie Groups

2.2. Interior Semi-Direct Product

2.3. Exterior Semi-Direct Product

3. Lie Group Actions

3.1. Group Actions

3.2. Equivariant Mappings

3.3. Orbit Space

3.4. Orbit Manifold

3.5. Orbit Reduction

3.6. Fundamental Vector Fields

4. Principal Bundles

4.1. Principal Group Actions

4.2. Quotient Manifolds

4.3. Morphisms of Principal Bundles

5. Associated Bundles

5.1. Frame Mappings

5.2. Associated Bundles

5.3. Morphisms of Associated Bundles

Notes and Additional Topics

Chapter 4: Lagrange Structures

1. Jet Prolongations of Fibered Manifolds

1.1. Jet Prolongations of Fibered Manifolds

1.2. Jet Prolongations of Morphisms

1.3. Horizontalization of Tangent Vectors

2. Trace Decompositions of Tensor Spaces

2.1. Trace Mappings

2.2. Kronecker Tensors

2.3. Duality in Tensor Spaces

2.4. Trace Decomposition

3. Differential Forms on Jet Prolongations of Fibered Manifolds

3.1. Horizontalization of Forms

3.2. Formal Derivative of a Function

3.3. The contact ideal

3.4. First Canonical Decomposition of Forms

3.5. Second Canonical Decomposition of Forms

3.6. Third Canonical Decomposition of Forms

3.7. The Poincare Lemma

4. Lagrangians, Variational Functionals

4.1. Fundamental Forms, Lagrangians

4.2. Lepage Forms

4.3. First Variation Formula

4.4. Extremals

4.5. The Euler-Lagrange Mapping

4.6. Higher Variations

5. Invariant Variational Principles and Conservation Laws

5.1. Invariant Lagrangians

5.2. Invariant Euler-Lagrange Forms

5.3. Symmetries of Extremals

6. Regularity and Hamilton Equations

6.1. Non-holonomic Jet Spaces

6.2. Hamilton Equations

6.3. Legendre Coordinates

6.4. Regular Variational Principles

Notes and Additional Topics

Chapter 5: Elementary Sheaf Theory

1. Sheaves

1.1. Local Homeomorphisms

1.2. Products and Fiber Products

1.3. Sheaf Spaces

1.4. The Quotient Sheaf Space

1.5. Prolongations of Continuous Sections

1.6. Sheaves

1.7. Presheaves

1.8. Complete Presheaves

1.9. Sheaf Associated with a Presheaf

2. Sheaf Cohomology

2.1. Exact Sequences of Abelian Groups

2.2. Complexes

2.3. The Connecting Morphism

2.4. Exact Sequences of Sheaves

2.5. The Canonical Resolution of a Sheaf

2.6. Cohomology Groups of a Sheaf

3. Sheaves over Paracompact Hausdorff Spaces

3.1. Soft Sheaves

3.2. Fine Sheaves

3.3. The Connecting Morphism

3.4. Acyclic Resolutions

Notes and Additional Topics

Chapter 6: Variational Sequences on Fibered Manifolds

1. The Variational Sequence

1.1. DeRham Sequence on Jet Bundles

1.2. Contact Subsequence

1.3. The Variational Sequence

1.4. Cohomology of the Variational Sequence

1.5. Interior Euler-Lagrange Operator

1.6. Lepage forms

2. The Structure of the Euler-Lagrange Mapping

2.1. Trivial Lagrangians

2.2. Dynamical Forms

2.3. The Inverse Problem of the Calculus of Variations

2.4. Local Variational Principles, First Variation Formula

2.5. Energy-Momentum Tensors for Differential Equations

2.6. Order Reducibility of Lagrangians

3. Variational Sequence and Symmetries

3.1. Lie Derivatives of Classes

3.2. Invariance of the Helmholtz Class

Notes and Additional Topics

Chapter 7: Invariant Variational Functionals on Principal Bundles

1. Prolongations of Principal Lie Group Actions

1.1. Jet Prolongations of a Principal Bundle

1.2. Bundles of Connections

1.3. Invariant Differential Forms

2. Invariant Variational Principles

2.1. Invariant Differential Forms

2.2. Invariant Vector Fields

3. The First Variation Formula

3.1. Euler-Lagrange Form

3.2. Noether’s Currents

4. Euler-Lagrange Equations

4.1. Extremals and Conservation Laws

Notes and Additional Topics

Chapter 8: Differential Invariants

1. Differential Invariants

1.1. Differential Groups

1.2. Normal Subgroup Structure

1.3. Differential Invariants

1.4. Orbit Reduction

1.5. Orbit Manifolds

2. Invariant Tensors

3.1. Absolute Invariant Tensors

3.2. Relative Invariant Tensors

3.3. Multilinear Invariants

3. Realizations of Differential Invariants

3.1. Higher Order Frame Bundles

3.2. Prolongations of Associated Bundles

3.3. Associated Lifting Functors

3.4. Natural Transformations

3.5. Differential Invariants and Lie Derivatives

Notes and Additional Topics

Chapter 9: Natural Variational Principles

1. Odd Base Forms on Fibered Manifolds

1.1. Orientation of a Vector Space and Odd Scalars

1.2. Orientation of a Manifold

1.3. Odd Scalars and Odd Forms on a Manifold

1.4. Odd Base Scalars, Odd Base Forms

1.5. The Pull-Back

1.6. Exterior Product of a Form and an Odd Base Form

1.7. Contraction by a Vector Field

1.8. Exterior Derivative

1.9. The Lie Derivative

2. Jet Prolongations of Frame Bundles

2.1. Prolongations of Group Action

2.2. Invariant Differential Forms

3. Natural Variational Principles on Frame Bundles

3.1. Invariant Differential Forms

3.2. Noether Currents

3.3. Euler-Lagrange Equations

4. Invariant Variational Principles on Associated Bundles

4.1. Tensor Bundles

4.2. First Variation Formula

4.3. Week Conservation Laws

4.4. Generalized Bianchi Identities

4.5. Euler-Lagrange Equations and Conservation Laws

5. Invariant Variational Principles in Tensor Bundles

5.1. Differential Invariants of a Metric Field

5.2. Invariant Variational Principles For the Metric Field

5.3. Differential Invariants of a Linear Connection

5.4. Invariant Variational Principles for Linear Connections

6. The Energy Functional and Harmonic Mappings

6.1. Energy of an Immersion

6.2. Extremals of the Energy Functional

Notes and Additional Topics

Appendices

1. Multi-Linear Algebra

2. Topology

3. Analysis

4. Categories

Bibliography

Index

## Description

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

## Readership

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.

## Details

- No. of pages:
- 500

- Language:
- English

- Copyright:
- © Elsevier Science 2000

- Published:
- 1st January 1980

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780080954288

## Ratings and Reviews

## About the Author

### Demeter Krupka

### Affiliations and Expertise

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