One of the purposes of this chapter is to discuss Hopf and unit Killing vector fields in the context of the theory of harmonic vector fields on Riemannian manifolds. The two classes of vector fields are related (cf.
Theorem 3.5
). Also unit Killing vector fields on Einstein manifolds are harmonic (cf.
Proposition 3.7
). More can be said on unit Killing vector fields on real space forms
M
n
(
c
) of (constant) sectional curvature
c
≥ 0. Indeed these are harmonic maps (cf.
Theorem 3.9
) and are actually parallel when
c
= 0. Starting from the result by S.D. Han & J.W. Yim,
[157]
(that the only harmonic vector fields on a sphere
S
3
are the Hopf vector fields, cf.
Theorem 3.10
) we report on recent findings by D. Perrone (cf.
[244]
) on ...
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