Harmonic Vector Fields

Chapter One. Geometry of the Tangent Bundle

Contents

1.1 The Tangent Bundle 2

1.2 Connections and Horizontal Vector Fields 4

1.3 The Dombrowski Map and the Sasaki Metric 6

1.3.1 Preliminaries on Local Calculations 9

1.3.2 Isotropic Almost Complex Structures 11

1.3.3 Invariant Isotropic Complex Structures 20

1.4 The Tangent Sphere Bundle 26

1.5 The Tangent Sphere Bundle over a Torus 29

The scope of Chapter 1 is to briefly review the basic facts in the geometry of the tangent bundle over a Riemannian manifold , such as nonlinear connections, the Dombrowski ...