The energy of unit vector fields on the 3-sphere

Details The energy of unit vector fields on the 3-sphere The energy of unit vector fields on the 3-sphere

2000-05-31

arxiv.org/abs/math/0005293v1

Mathematics

Differential Geometry

AMSTeX, 20 pages, to appear in J Geom Phys

Scientific paper

10.1016/S0393-0440(00)00034-6

The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of the 3-dimensional Hopf vector field is compared and contrasted with that of the closely-related Hopf map. Finally, it is shown that the Hopf vector fields are the unique global minima of the energy functional restricted to unit vector fields on the 3-sphere.

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