Harmonic Vector Fields

by Dragomir; Perrone

• ISBN13: 9780124158269
• ISBN10: 0124158269
• Format: Hardcover
• Copyright: 2011-10-26
• Publisher: Elsevier Science

Summary

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

Preface	p. ix
Geometry of the Tangent Bundle	p. 1
The Tangent Bundle	p. 2
Connections and Horizontal Vector Fields	p. 4
The Dombrowski Map and the Sasaki Metric	p. 6
The Tangent Sphere Bundle	p. 26
The Tangent Sphere Bundle over a Torus	p. 29
Harmonic Vector Fields	p. 37
Vector Fields as Isometric Immersions	p. 38
The Energy of a Vector Field	p. 41
Vector Fields Which Are Harmonic Maps	p. 46
The Tension of a Vector Field	p. 49
Variations through Vector Fields	p. 56
Unit Vector Fields	p. 58
The Second Variation of the Energy Function	p. 73
Unboundedness of the Energy Functional	p. 81
The Dirichlet Problem	p. 82
Conformal Change of Metric on the Torus	p. 106
Sobolev Spaces of Vector Fields	p. 108
Harmonicity and Stability	p. 129
Hopf Vector Fields on Spheres	p. 130
The Energy of Unit Killing Fields in Dimension 3	p. 140
instability of Hopf Vector Fields	p. 146
Existence of Minima in Dimension > 3	p. 151
Brito's Functional	p. 155
The Brito Energy of the Reeb Vector	p. 158
Vector Fields with Singularities	p. 164
Normal Vector Fields on Principal Orbits	p. 179
RiemannianTori	p. 188
Harmonicity and Contact Metric Structures	p. 205
H-Contact Manifolds	p. 206
Three-Dimensional H-Contact Manifolds	p. 218
Stability of the Reeb Vector Field	p. 233
Harmonic Almost Contact Structures	p. 243
Reeb Vector Fields on Real Hypersurfaces	p. 245
Harmonicity and Stability of the Geodesic Flow	p. 259
Harmonicity with Respect to g-Natural Metrics	p. 273
g-Natural Metrics	p. 275
Naturally Harmonic Vector Fields	p. 282
Vector Fields Which Are Naturally Harmonic Maps	p. 290
Geodesic Flow with Respect to g-Natural Metrics	p. 302
The Energy of Sections	p. 307
The Horizontal Bundle	p. 309
The Sasaki Metric	p. 316
The Sphere Bundle U(E)	p. 320
The Energy of Cross Sections	p. 324
Unit Sections	p. 326
Harmonic Sections in Normal Bundles	p. 329
The Energy of Oriented Distributions	p. 332
Examples of Harmonic Distributions	p. 337
The Chacon-Naveira Energy	p. 344
Harmonic Vector Fields in CR Geometry	p. 355
The Canonical Metric	p. 359
Bundles of Hyperquadrics in (T(M),J, Gs)	p. 365
Harmonic Vector Fields from C(M)	p. 377
Boundary Values of Bergman-Harmonic Maps	p. 387
Pseudo harmonic Maps	p. 389
The Pseudo hermitian Biegung	p. 394
The Second Variation Formula	p. 401
Lorentz Geometry and Harmonic Vector Fields	p. 407
A Few Notions of Lorentz Geometry	p. 407
Energy Functionals and Tension Fields	p. 410
The Spacelike Energy	p. 412
The Second Variation of the Spacelike Energy	p. 431
Conformal Vector Fields	p. 434
Twisted Cohomologies	p. 437
The Stokes Theorem on Complete Manifolds	p. 447
Complex Monge-Ampere Equations	p. 457
Introduction	p. 457
Strictly Parabolic Manifolds	p. 460
Foliations and Monge-Ampere Equations	p. 461
Adapted Complex Structures	p. 464
CR Submanifolds of Grauert Tubes	p. 468
Exceptional Orbits of Highest Dimension	p. 473
Reilly's Formula	p. 479
References	p. 491
Index	p. 505
Table of Contents provided by Ingram. All Rights Reserved.