Harmonic Vector Fields

Chapter Five. Harmonicity with Respect to g -Natural Metrics

Contents

5.1 g -Natural Metrics 275

5.1.1 Generalized Cheeger-Gromoll Metrics 279

5.1.2 g -Natural Riemannian Metrics on S_ρ( M) 280

5.2 Naturally Harmonic Vector Fields 282

5.2.1 The Energy of V : ( M, g ) → ( T( M), g ) 282

5.2.2 The Tension Field of V : ( M, g ) → ( T( M), g ) 284

5.2.3 Naturally Harmonic Vector Fields 286

5.2.4 Vertically Harmonic Vector Fields 288

5.2.5 Naturally Harmonic Unit Vector Fields 289

5.3 Vector Fields Which Are Naturally Harmonic Maps 290

5.3.1 A Generalization of the Ishihara-Nouhaud Theorem 291

5.3.2 Non-parallel Vector Fields Which Are Harmonic Maps 293

5.3.3 Unit Vector Fields Which Are Harmonic Maps 295

5.4 Geodesic Flow with Respect to g -Natural Metrics 302 ...