Chapter One - Geometry of the Tangent Bundle

Abstract

This chapter briefly reviews the basic facts in the geometry of the tangent bundle over a Riemannian manifold, such as nonlinear connections, the Dombrowski map, and the Sasaki metric. Tangent bundle also carries a natural almost complex structure compatible to Sasaki metric and such that tangent bundle and complex structure are almost Kahler manifold. The results offer the possibility of further development for the harmonic vector field theory. A deep circle of ideas relates the geometry of the tangent bundle over a Riemannian manifold to the study of the global solutions to the homogeneous complex Monge-Ampere equation. It further examines the behavior of the total bending functional under conformational transformation. Conclusive and complete results on harmonic vector fields on Riemannian tori are presented.