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On the energy of a unit vector field. (English) Zbl 0878.58017
Let \(M\) be a closed oriented \(m\)-dimensional Riemannian manifold, \(T_1M\) its unit tangent bundle equipped with the restriction of the Sasaki metric on \(TM\), and \(U\) a unit vector field on \(M\).
The author considers the energy of \(U\) viewed as a mapping \(U:M\to T_1M\) between Riemannian manifolds. The constrained variational problem is studied, where variations are confined to unit vector fields, and the first and second variational formulas are derived. The Hopf vector fields on \(M= S^{2n+1}\), that is the unit vector fields tangent to the fibres of the Hopf fibration, or any congruent foliation, are shown to be critical points of the energy functional subject to the constraints. If \(U\) is such a vector field, then \(U\) is unstable for \(n\geq 2\), with an energy index (relative to the constraint manifold) of at least \(2n+2\).

MSC:
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
53C20 Global Riemannian geometry, including pinching
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58E20 Harmonic maps, etc.
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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