November 2011
Intermediate to advanced
528 pages
10h 30m
English
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One purpose of Chapter 3 is to discuss Hopf and unit Killing vector fields in the context of the theory of harmonic vector fields on Riemannian manifolds. Also unit Killing vector fields on Einstein manifolds are harmonic. 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 and are actually parallel when
c
= 0. Starting from the result by S.D. Han & J.W. Yim (1998) we report on recent findings by D. Perrone (2008) on unit Killing vector fields on 3-dimensional Riemannian manifolds. Motivated by a result of Y.L. Xin (1980) in harmonic map theory one expects instability of Hopf vector fields on spheres. As it turns out, this is indeed the case for ...
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