Harmonic maps and sections on spheres

Details Harmonic maps and sections on spheres Harmonic maps and sections on spheres

2007-03-02

arxiv.org/abs/math/0703060v1

Mathematics

Differential Geometry

A detailed version is available at: http://maths2.univ-brest.fr/~loubeau/articles/harm-long.pdf

Scientific paper

The absence of interesting harmonic sections for the Sasaki and Cheeger-Gromoll metrics has led to the consideration of alternatives, for example in the form of a two-parameter family of natural metrics shown to relax existence conditions for harmonicity. This article investigates harmonic Killing vector fields, proves their non-existence on S^2, obtains rigidity results for harmonic gradient vector fields on the two-sphere, classifies spherical quadratic gradient fields in all dimensions and determines the tension field, concluding with the discovery of a family of metrics making Hopf vector fields harmonic maps on S^{2n+1}.

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