Harmonic vector fields on Riemannian manifolds

This dissertation investigates harmonic vector fields which are special mappings on Riemannian manifolds with many interesting properties. It aims for a sharp definition of these fields through a focus on several aspects of geometry. Key concepts include the Weitzenböck formula, the divergence theorem, the Euler-Lagrange equation and the Sasaki metric. This particular metric contains horizontal and vertical components which are used to define vertical energy and this, in turn, leads to a definition of harmonic vector fields which are later generalised by the Cheeger-Gromoll metric and the general definition of a harmonic vector field. The dissertation also concentrates on some specific examples of harmonic vector fields such as harmonic unit vector fields, the Hopf vector field, conformal gradient fields on the unit sphere and on the hyperbolic space. The key outcome of this research, presented in the concluding subsections of the dissertation, is the discovery of two new examples that give fresh insight into this important aspect of differential geometry.