Harmonic Vector Fields

Variational Principles and Differential Geometry

Sorin Dragomir, Domenico Perrone

• Publisher: Elsevier
• Year: 2011
• Print ISBN:
  9780124158269
• eBook ISBN:
  9780124160323
• Topic: Physical Sciences
• Subtopic: Mathematical Analysis

An excellent reference for anyone needing to examine properties of harmonic vector fields to help them solve research problems. The book provides the main results of harmonic vector fields with an emphasis on Riemannian manifolds using past and existing problems to assist you in analyzing and furnishing your own conclusion for further research. It emphasizes a combination of theoretical development with practical applications for a solid treatment of the subject useful to those new to research using differential geometric methods in extensive detail. - 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

Table of contents

• Cover image
• Table of Contents
• Front Matter
• Copyright
• Preface
• Chapter One. Geometry of the Tangent Bundle
• 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
• Chapter Six. The Energy of Sections
• Chapter Seven. Harmonic Vector Fields in CR Geometry
• Chapter Eight. Lorentz Geometry and Harmonic 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