Harmonic vector fields variational principles and differential geometry
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 ?elds with an emphasis on Riemannian manifolds using past and existing problems to assist you in analyzing and furnis...
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| Formato: | Libro electrónico |
| Idioma: | Inglés |
| Publicado: |
New York :
Academic Press
2012.
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| Edición: | 1st edition |
| Materias: | |
| Ver en Biblioteca Universitat Ramon Llull: | https://discovery.url.edu/permalink/34CSUC_URL/1im36ta/alma991009628243606719 |
Tabla de Contenidos:
- Front Cover; Harmonic Vector Fields: Variational Principles and Differential Geometry; Copyright; Table of Contents; Preface; Chapter 1 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; 1.3.1. Preliminaries on Local Calculations; 1.3.2. Isotropic Almost Complex Structures; 1.3.3. Invariant Isotropic Complex Structures; 1.4. The Tangent Sphere Bundle; 1.5. The Tangent Sphere Bundle over a Torus; Chapter 2 Harmonic Vector Fields; 2.1. Vector Fields as Isometric Immersions
- 2.2. The Energy of a Vector Field2.3. 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; 2.9.1. The Graham-Lee Connection; 2.9.2. The Levi-Civita Connection of the Bergman Metric; 2.9.3. Proof of Theorem 2.36; 2.9.4. Proof of Theorem 2.37; 2.9.5. Final Comments; 2.10. Conformal Change of Metric on the Torus; 2.11. Sobolev Spaces of Vector Fields; Chapter 3 Harmonicity and Stability
- 3.1. Hopf Vector Fields on Spheres3.2. 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; 3.7.1. Geodesic Distance; 3.7.2. F. Brito & P.G. Walczak's Theorem; 3.7.3. Harmonic Radial Vector Fields; 3.8. Normal Vector Fields on Principal Orbits; 3.9. Riemannian Tori; 3.9.1. Harmonic Vector Fields on Riemannian Tori; 3.9.2. Stability; 3.9.3. Examples and Open Problems
- Chapter 4 Harmonicity and Contact Metric Structures4.1. H-Contact Manifolds; 4.1.1. Contact Metric Manifolds; 4.1.2. H-Contact Manifolds; 4.2. Three-Dimensional H-Contact Manifolds; 4.2.1. A Characterization of H-Contact Three-Manifolds and New Examples; 4.2.2. Taut Contact Circles and H-Contact Structures; 4.3. Stability of the Reeb Vector Field; 4.3.1. Stability of ? for Sasakian 3-Manifolds and Generalized (k,μ)-Spaces; 4.3.2. Stability of Strongly Normal Reeb Vector Fields; 4.4. Harmonic Almost Contact Structures; 4.5. Reeb Vector Fields on Real Hypersurfaces
- 4.5.1. The Rough Laplacian and Criteria of Harmonicity4.5.2. Ruled Hypersurfaces; 4.5.3. Real Hypersurfaces of Contact Type; 4.6. Harmonicity and Stability of the Geodesic Flow; 4.6.1. The Ricci Curvature; 4.6.2. H-Contact Tangent Sphere Bundles; 4.6.3. The Stability of the Geodesic Flow; Chapter 5 Harmonicity with Respect to g-Natural Metrics; 5.1. g-Natural Metrics; 5.1.1. Generalized Cheeger-Gromoll Metrics; 5.1.2. g-Natural Riemannian Metrics on S?(M); 5.2. Naturally Harmonic Vector Fields; 5.2.1. The Energy of V : (M,g) -> (T(M), G); 5.2.2. The Tension Field of V : (M,g) -> (T(M),G)
- 5.2.3. Naturally Harmonic Vector Fields
