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.

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


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

Table of Contents


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


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"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