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.
1. Cover image 2. Table of Contents 3. Front Matter 4. Copyright 5. Preface 6. Chapter One. Geometry of the Tangent Bundle 7. 1.1. The Tangent Bundle 8. 1.2. Connections and Horizontal Vector Fields 9. 1.3. The Dombrowski Map and the Sasaki Metric 10. 1.4. The Tangent Sphere Bundle 11. 1.5. The Tangent Sphere Bundle over a Torus 12. Chapter Two. Harmonic Vector Fields 13. 2.1. Vector Fields as Isometric Immersions 14. 2.2. The Energy of a Vector Field 15. 2.3. Vector Fields Which Are Harmonic Maps 16. 2.4. The Tension of a Vector Field 17. 2.5. Variations through Vector Fields 18. 2.6. Unit Vector Fields 19. 2.7. The Second Variation of the Energy Function 20. 2.8. Unboundedness of the Energy Functional 21. 2.9. The Dirichlet Problem 22. 2.10. Conformal Change of Metric on the Torus 23. 2.11. Sobolev Spaces of Vector Fields 24. Chapter Three. Harmonicity and Stability 25. 3.1. Hopf Vector Fields on Spheres 26. 3.2. The Energy of Unit Killing Fields in Dimension 3 27. 3.3. Instability of Hopf Vector Fields 28. 3.4. Existence of Minima in Dimension > 3 29. 3.5. Brito's Functional 30. 3.6. The Brito Energy of the Reeb Vector 31. 3.7. Vector Fields with Singularities 32. 3.8. Normal Vector Fields on Principal Orbits 33. 3.9. Riemannian Tori 34. Chapter Four. Harmonicity and Contact Metric Structures 35. 4.1. H-Contact Manifolds 36. 4.2. Three-Dimensional H-Contact Manifolds 37. 4.3. Stability of the Reeb Vector Field 38. 4.4. Harmonic Almost Contact Structures 39. 4.5. Reeb Vector Fields on Real Hypersurfaces 40. 4.6. Harmonicity and Stability of the Geodesic Flow 41. Chapter Five. Harmonicity with Respect to g-Natural Metrics 42. 5.1. g-Natural Metrics 43. 5.2. Naturally Harmonic Vector Fields 44. 5.3. Vector Fields Which Are Naturally Harmonic Maps 45. 5.4. Geodesic Flow with Respect to g-Natural Metrics 46. Chapter Six. The Energy of Sections 47. 6.1. The Horizontal Bundle 48. 6.2. The Sasaki Metric 49. 6.3. The Sphere Bundle U(E) 50. 6.4. The Energy of Cross Sections 51. 6.5. Unit Sections 52. 6.6. Harmonic Sections in Normal Bundles 53. 6.7. The Energy of Oriented Distributions 54. 6.8. Examples of Harmonic Distributions 55. 6.9. The Chacon-Naveira Energy 56. Chapter Seven. Harmonic Vector Fields in CR Geometry 57. 7.1. The Canonical Metric 58. 7.2. Bundles of Hyperquadrics in (T(M), J, Gs) 59. 7.3. Harmonic Vector Fields from C(M) 60. 7.4. Boundary Values of Bergman-Harmonic Maps 61. 7.5. Pseudoharmonic Maps 62. 7.6. The Pseudohermitian Biegung 63. 7.7. The Second Variation Formula 64. Chapter Eight. Lorentz Geometry and Harmonic Vector Fields 65. 8.1. A Few Notions of Lorentz Geometry 66. 8.2. Energy Functionals and Tension Fields 67. 8.3. The Spacelike Energy 68. 8.4. The Second Variation of the Spacelike Energy 69. 8.5. Conformal Vector Fields 70. Appendix A. Twisted Cohomologies 71. Appendix B. The Stokes Theorem on Complete Manifolds 72. Appendix C. Complex Monge-Ampère Equations 73. Appendix D. Exceptional Orbits of Highest Dimension 74. Appendix E. Reilly's Formula 75. References 76. Index
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.
1. Cover image 2. Table of Contents 3. Front Matter 4. Copyright 5. Preface 6. Chapter One. Geometry of the Tangent Bundle 7. 1.1. The Tangent Bundle 8. 1.2. Connections and Horizontal Vector Fields 9. 1.3. The Dombrowski Map and the Sasaki Metric 10. 1.4. The Tangent Sphere Bundle 11. 1.5. The Tangent Sphere Bundle over a Torus 12. Chapter Two. Harmonic Vector Fields 13. 2.1. Vector Fields as Isometric Immersions 14. 2.2. The Energy of a Vector Field 15. 2.3. Vector Fields Which Are Harmonic Maps 16. 2.4. The Tension of a Vector Field 17. 2.5. Variations through Vector Fields 18. 2.6. Unit Vector Fields 19. 2.7. The Second Variation of the Energy Function 20. 2.8. Unboundedness of the Energy Functional 21. 2.9. The Dirichlet Problem 22. 2.10. Conformal Change of Metric on the Torus 23. 2.11. Sobolev Spaces of Vector Fields 24. Chapter Three. Harmonicity and Stability 25. 3.1. Hopf Vector Fields on Spheres 26. 3.2. The Energy of Unit Killing Fields in Dimension 3 27. 3.3. Instability of Hopf Vector Fields 28. 3.4. Existence of Minima in Dimension > 3 29. 3.5. Brito's Functional 30. 3.6. The Brito Energy of the Reeb Vector 31. 3.7. Vector Fields with Singularities 32. 3.8. Normal Vector Fields on Principal Orbits 33. 3.9. Riemannian Tori 34. Chapter Four. Harmonicity and Contact Metric Structures 35. 4.1. H-Contact Manifolds 36. 4.2. Three-Dimensional H-Contact Manifolds 37. 4.3. Stability of the Reeb Vector Field 38. 4.4. Harmonic Almost Contact Structures 39. 4.5. Reeb Vector Fields on Real Hypersurfaces 40. 4.6. Harmonicity and Stability of the Geodesic Flow 41. Chapter Five. Harmonicity with Respect to g-Natural Metrics 42. 5.1. g-Natural Metrics 43. 5.2. Naturally Harmonic Vector Fields 44. 5.3. Vector Fields Which Are Naturally Harmonic Maps 45. 5.4. Geodesic Flow with Respect to g-Natural Metrics 46. Chapter Six. The Energy of Sections 47. 6.1. The Horizontal Bundle 48. 6.2. The Sasaki Metric 49. 6.3. The Sphere Bundle U(E) 50. 6.4. The Energy of Cross Sections 51. 6.5. Unit Sections 52. 6.6. Harmonic Sections in Normal Bundles 53. 6.7. The Energy of Oriented Distributions 54. 6.8. Examples of Harmonic Distributions 55. 6.9. The Chacon-Naveira Energy 56. Chapter Seven. Harmonic Vector Fields in CR Geometry 57. 7.1. The Canonical Metric 58. 7.2. Bundles of Hyperquadrics in (T(M), J, Gs) 59. 7.3. Harmonic Vector Fields from C(M) 60. 7.4. Boundary Values of Bergman-Harmonic Maps 61. 7.5. Pseudoharmonic Maps 62. 7.6. The Pseudohermitian Biegung 63. 7.7. The Second Variation Formula 64. Chapter Eight. Lorentz Geometry and Harmonic Vector Fields 65. 8.1. A Few Notions of Lorentz Geometry 66. 8.2. Energy Functionals and Tension Fields 67. 8.3. The Spacelike Energy 68. 8.4. The Second Variation of the Spacelike Energy 69. 8.5. Conformal Vector Fields 70. Appendix A. Twisted Cohomologies 71. Appendix B. The Stokes Theorem on Complete Manifolds 72. Appendix C. Complex Monge-Ampère Equations 73. Appendix D. Exceptional Orbits of Highest Dimension 74. Appendix E. Reilly's Formula 75. References 76. Index
