# Harmonic Vector Fields

## 1st Edition

### Variational Principles and Differential Geometry

**Authors:**Sorin Dragomir Domenico Perrone

**Hardcover ISBN:**9780124158269

**eBook ISBN:**9780124160323

**Imprint:**Elsevier

**Published Date:**26th October 2011

**Page Count:**528

## Description

Preface

Chapter One. Geometry of the Tangent Bundle

1.1. The Tangent Bundle

1.2. Connections and Horizontal Vector Fields

1.3. The Dombrowski Map and the Sasaki Metric

1.4. The Tangent Sphere Bundle

1.5. The Tangent Sphere Bundle over a Torus

Chapter Two. Harmonic Vector Fields

2.1. Vector Fields as Isometric Immersions

2.2. The Energy of a Vector Field

2.3. Vector Fields Which Are Harmonic Maps

2.4. The Tension of a Vector Field

2.5. Variations through Vector Fields

2.6. Unit Vector Fields

2.7. The Second Variation of the Energy Function

2.8. Unboundedness of the Energy Functional

2.9. The Dirichlet Problem

2.10. Conformal Change of Metric on the Torus

2.11. Sobolev Spaces of Vector Fields

Chapter Three. Harmonicity and Stability

3.1. Hopf Vector Fields on Spheres

3.2. The Energy of Unit Killing Fields in Dimension 3

3.3. Instability of Hopf Vector Fields

3.4. Existence of Minima in Dimension > 3

3.5. Brito's Functional

3.6. The Brito Energy of the Reeb Vector

3.7. Vector Fields with Singularities

3.8. Normal Vector Fields on Principal Orbits

3.9. Riemannian Tori

Chapter Four. Harmonicity and Contact Metric Structures

4.1. H-Contact Manifolds

4.2. Three-Dimensional H-Contact Manifolds

4.3. Stability of the Reeb Vector Field

4.4. Harmonic Almost Contact Structures

4.5. Reeb Vector Fields on Real Hypersurfaces

4.6. Harmonicity and Stability of the Geodesic Flow

Chapter Five. Harmonicity with Respect to g-Natural Metrics

5.1. g-Natural Metrics

5.2. Naturally Harmonic Vector Fields

5.3. Vector Fields Which Are Naturally Harmonic Maps

5.4. Geodesic Flow with Respect to g-Natural Metrics

Chapter Six. The Energy of Sections

6.1. The Horizontal Bundle

6.2. The Sasaki Metric

6.3. The Sphere Bundle U(E)

6.4. The Energy of Cross Sections

6.5. Unit Sections

6.6. Harmonic Sections in Normal Bundles

6.7. The Energy of Oriented Distributions

6.8. Examples of Harmonic Distributions

6.9. The Chacon-Naveira Energy

Chapter Seven. Harmonic Vector Fields in CR Geometry

7.1. The Canonical Metric

7.2. Bundles of Hyperquadrics in (T(M), J, Gs)

7.3. Harmonic Vector Fields from C(M)

7.4. Boundary Values of Bergman-Harmonic Maps

7.5. Pseudoharmonic Maps

7.6. The Pseudohermitian Biegung

7.7. The Second Variation Formula

Chapter Eight. Lorentz Geometry and Harmonic Vector Fields

8.1. A Few Notions of Lorentz Geometry

8.2. Energy Functionals and Tension Fields

8.3. The Spacelike Energy

8.4. The Second Variation of the Spacelike Energy

8.5. Conformal Vector Fields

Appendix A. Twisted Cohomologies

Appendix B. The Stokes Theorem on Complete Manifolds

Appendix C. Complex Monge-Ampère Equations

Appendix D. Exceptional Orbits of Highest Dimension

Appendix E. Reilly's Formula

References

Index

## Key Features

- A useful tool for any scientist conducting research in the field of harmonic analysis
- Provides applications and modern techniques to problem solving
- A clear and concise exposition of differential geometry of harmonic vector fields on Reimannian manifolds
- Physical Applications of Geometric Methods

## Readership

Computer & Physical Scientists, Engineers, Applied Mathematicians, Structural Geologists

## Table of Contents

Preface

Chapter One. Geometry of the Tangent Bundle

1.1. The Tangent Bundle

1.2. Connections and Horizontal Vector Fields

1.3. The Dombrowski Map and the Sasaki Metric

1.4. The Tangent Sphere Bundle

1.5. The Tangent Sphere Bundle over a Torus

Chapter Two. Harmonic Vector Fields

2.1. Vector Fields as Isometric Immersions

2.2. The Energy of a Vector Field

2.3. Vector Fields Which Are Harmonic Maps

2.4. The Tension of a Vector Field

2.5. Variations through Vector Fields

2.6. Unit Vector Fields

2.7. The Second Variation of the Energy Function

2.8. Unboundedness of the Energy Functional

2.9. The Dirichlet Problem

2.10. Conformal Change of Metric on the Torus

2.11. Sobolev Spaces of Vector Fields

Chapter Three. Harmonicity and Stability

3.1. Hopf Vector Fields on Spheres

3.2. The Energy of Unit Killing Fields in Dimension 3

3.3. Instability of Hopf Vector Fields

3.4. Existence of Minima in Dimension > 3

3.5. Brito's Functional

3.6. The Brito Energy of the Reeb Vector

3.7. Vector Fields with Singularities

3.8. Normal Vector Fields on Principal Orbits

3.9. Riemannian Tori

Chapter Four. Harmonicity and Contact Metric Structures

4.1. H-Contact Manifolds

4.2. Three-Dimensional H-Contact Manifolds

4.3. Stability of the Reeb Vector Field

4.4. Harmonic Almost Contact Structures

4.5. Reeb Vector Fields on Real Hypersurfaces

4.6. Harmonicity and Stability of the Geodesic Flow

Chapter Five. Harmonicity with Respect to g-Natural Metrics

5.1. g-Natural Metrics

5.2. Naturally Harmonic Vector Fields

5.3. Vector Fields Which Are Naturally Harmonic Maps

5.4. Geodesic Flow with Respect to g-Natural Metrics

Chapter Six. The Energy of Sections

6.1. The Horizontal Bundle

6.2. The Sasaki Metric

6.3. The Sphere Bundle U(E)

6.4. The Energy of Cross Sections

6.5. Unit Sections

6.6. Harmonic Sections in Normal Bundles

6.7. The Energy of Oriented Distributions

6.8. Examples of Harmonic Distributions

6.9. The Chacon-Naveira Energy

Chapter Seven. Harmonic Vector Fields in CR Geometry

7.1. The Canonical Metric

7.2. Bundles of Hyperquadrics in (T(M), J, Gs)

7.3. Harmonic Vector Fields from C(M)

7.4. Boundary Values of Bergman-Harmonic Maps

7.5. Pseudoharmonic Maps

7.6. The Pseudohermitian Biegung

7.7. The Second Variation Formula

Chapter Eight. Lorentz Geometry and Harmonic Vector Fields

8.1. A Few Notions of Lorentz Geometry

8.2. Energy Functionals and Tension Fields

8.3. The Spacelike Energy

8.4. The Second Variation of the Spacelike Energy

8.5. Conformal Vector Fields

Appendix A. Twisted Cohomologies

Appendix B. The Stokes Theorem on Complete Manifolds

Appendix C. Complex Monge-Ampère Equations

Appendix D. Exceptional Orbits of Highest Dimension

Appendix E. Reilly's Formula

References

Index

## Details

- No. of pages:
- 528

- Language:
- English

- Copyright:
- © Elsevier 2011

- Published:
- 26th October 2011

- Imprint:
- Elsevier

- eBook ISBN:
- 9780124160323

- Hardcover ISBN:
- 9780124158269

## About the Author

### Sorin Dragomir

### Affiliations and Expertise

University of Basilicata, Potenza, Italy

### Domenico Perrone

### Affiliations and Expertise

Universita' del Salento, Lecce, Italy

## Reviews

"This monograph (over 500 pages) is well written and self-contained in the field of harmonic vector fields..."--**Mathematical Reviews, Harmonic Vector Fields**

*"The book is certainly a valuable reference source…The bibliography appears both extensive and carefully selected...The style of formal statements is clear and helpful when browsing for specific results."--Zentralblatt MATH 2012-1245-53002*